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a sandbox at the beach, oil painting style

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Perfect Beach Scene: 7 Hours of White Sand, Blue Water & Ocean Waves in 4K

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In my life I have been fortunate to have explored fascinating places around the world.  There is one particular city though, that keeps giving me new emotions every day. It is Roma (or Rome).
Its variety of architectures that stands clearly apart from other "megacities", Its unparalleled history legacy and Its intimate and quieter sides make Rome a city that does get into your bloodstream. 

Watch my other works:
Cuba: https://vimeo.com/224883799
Venezia: https://vimeo.com/181612110
Lanzarote: https://vimeo.com/192406523
Reverie of Vietnam: https://vimeo.com/126747807
Moments of Puglia: https://vimeo.com/165102515
Malta: https://vimeo.com/202709486

Rome has been documented in different ways and for different purposes but few videos make you appreciate this city through local's eyes. 

I have dedicated months filming everyday's moment to describe a day in Rome through a roman's eyes. This short film is the result. Let me indulge you for a few minutes.  

Awards:
OFFICIAL SELECTION - 2016 ICFF – Italian Contemporary Film Festival

Locations:
Many filmed locations are unknown to many. Click on the link to see film locations on map: https://goo.gl/AWgM6g

Based on an original idea and directed by Oliver Astrologo - FB: http://fb.com/oliver.astrologo
Additional footage: Nils Astrologo

Original Score: 
Max LL - http://maxll.ca

The original soundtrack of ROMA has been produced by talented composer Max LL. All music and sounds effects rights are reserved. 

SteadyCam (Gimbal) support
Aerea Visus - Uniclamp : http://goo.gl/KMDqbs

Thanks:
Marco Baroni di Fonderia, Simone Sampò, Alice Celia, Federico Chericoni and everyone who helped me create this video.

Appearances:
Maurizio & Satia Martinoli, Alessia Scognamiglio, Andrea di Curcio, Stefano Silvani, Valeria D’Ovidio, Tommaso Cassinis. The mask: Jacopo Sabar Giacchino.

Flickr album: https://flic.kr/s/aHskitMuGV

Geek Stuff
I have received so many messages asking for details on how I produce my videos. 
I put down this short piece: http://oliverastrologo.com/videotips this explains few techniques I have been using and includes a list of equipment. Don't hesitate to write to me if you need further details or have any recommendation.

Gear (Find a comprensive list of my equipment here: http://oliverastrologo.com/gear)
Sony A7ii @ 1080 50p PP7 (Exposed in S-Log2)
Sony A7s @ 1080 50p PP7 (Exposed in S-Log2)

Video Stabilization: Area Visus AV-ST-01

Lens used:
Zeiss 35mm 2.8
Zeiss 55mm 1.8
Canon 24-15mm 4.0 IS

Color Correction
ImpulZ LUTs with some tweaks. 

Riprese aeree (in italiano):
La questione delle riprese aeree effettuate con Droni è spesso soggetta a controversie, è importante ricordare che possono essere (e spesso sono) utilizzati in modo responsabile. Le riprese aeree di questo video sono state realizzate prima del 30 Aprile 2014 data antecedente alle nuove restrizioni sui voli, tutte le immagini sono state acquisite rispettando le seguenti modalità:

- Evitare le zone sensibili o qualsiasi cosa che può essere frainteso senza previa autorizzazione.
- Evitare i periodi più affollati, le ore di punta, eventi speciali, ecc.
- Evitare le folle. In caso di zone ove erano presenti delle persone con osservatori di terra e ricetrasmittenti.
- Rispettare le aree residenziali di tranquillità e diritto alla privacy.
- Eliminare qualsiasi materiale che rivela inavvertitamente persone o che possa compromettere la privacy.

Per domande riguardanti le riprese aeree, vi prego di contattarmi via e-mail.

** Important **
All video, music and sounds effects are rights reserved.
Please only use and share this embed code of the original video.
Third party downloads and distribution is not permitted.

If you are interested in buying footage or any inquiry please contact: info@oliverastrologo.com
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Instagram: https://instagram.com/oliverhl
Twitter https://twitter.com/oliverastrologo
Facebook https://fb.com/oliver.astrologo
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When you see the crimson and orange tide sweep over tree-lined streets and leave a stunning foliage display in its wake, it's autumn! Unlike the scorching sun in summer or the bitter cold in winter, September welcomes the season with a cozy atmosphere and pleasant weather, setting the stage for a productive month ahead. But guess what? You can even double your productivity with a well-planned schedule. We've got you covered with our September 2024 calendar, where all the important events and schedules are on the same page. Let's see what we have on Wiki Calendar!

Printable September 2024 Calendar

Our September calendars let you view all 12 months of 2024 on one page, offering a clear annual overview. Easily switch between months with a single click. Beyond printing, download the calendar as a .pdf file. Add personal notes or events directly to the calendar for a more tailored experience. Integrate with popular calendar apps to seamlessly import and export events and reminders.

Why not a weekly or a yearly but a monthly template? While a 7-day calendar grants you more room for jotting down notes, it hinders you from viewing the monthly agenda as a whole. Meanwhile, a yearly template might leave you overwhelmed with tons of events and deadlines, making it hard to keep track of your schedule. And a September 2024 calendar template strikes a perfect balance between the two.

30 date boxes are generously designed for all your notes while still sitting neatly within the compact frame of US-letter size (8.5 x 11 inches). These dimensions grant enough room for your monthly schedule, which can be captured at a mere glance, without leaving a big footprint on your desk. Coming in standard PDF, JPG, PNG formats, these templates allow you to download and print them out in a breeze. No matter what type of device and printer you use, the quality of the printout remains unchanged, ensuring a seamless experience for all users. This printable calendar for September 2024 is also available in various themes, catering to different preferences and tastes. Curious about our meticulously crafted collection? All will be displayed below!

September 2024 Calendar Template

Drawing on our extensive expertise in calendar design, we present an array of September 2024 calendars, meticulously fashioned to cater to both digital and physical formats. These tools are engineered to refine your scheduling process, featuring an array of options such as week numbers and adaptable layouts that incorporate key U.S. holidays. Our online calendars for September offer a dynamic and interactive planning solution, equipped with customization capabilities for holiday settings and a variety of templates including blank, monthly, and weekly views, ideal for comprehensive planning over several years.

Our online September 2024 calendar enhances your planning with features like lunar phase tracking, day/night calculations, and personal event integration. It adapts to reflect international holidays, matching your specific location. Our printable calendars combine visual appeal with options for customization, perfect for personal projects or creative use. Whether you prefer the tactile experience of a printed calendar or the flexibility of digital planning, our September calendars are designed to meet your needs with style and efficiency.

Blank September Calendar 2024

Some folks pursue a minimalist lifestyle, while others love to adorn their workspace with vibrant and full-of-life elements. That's why we've created this blank September 2024 calendar to cater to you all. The black-and-white design appears clear and clutter-free, allowing you to jot down all the events and notes without overwhelming the background. This template also serves as a blank canvas for artist souls to customize their calendars. Adding some personal touches to the design will speak volumes about your personality and lifestyle while still leaving enough space for your schedule.

Each date box is crafted with crisp and clear numbers, along with a generous room for your notes. No more scattered sticky memos on your desk and computer! Now, you can capture all important events, meetings, and deadlines of the whole month at a mere glance. Print it out to organize your chock-full agenda!

Floral Calendar for September

Want a stunning and uplifting calendar but don't have good aesthetic taste? We've got you covered with this 2024 September floral calendar. This template bears a timeless beauty, evoking positive emotions and boosting your productivity. Various types and sizes of flowers are delicately woven into a charming canvas, setting the tone for the month ahead. The harmonious color palette exudes a calming sense, reducing your stress and anxiety when working or studying for a long time.

A quick glance at this floral oasis offers a soothing escape from the hustle and bustle of life. It exposes you to the comforting embrace of nature, unlocking your creativity and enhancing your focus on tasks. Don't worry that these decorative elements will draw you away from your work. Each flower is presented in a subtle tone, adding vibrancy to the background without overshadowing your notes. Download it, and make every day count!

September Calendar Monday Start

We've all been there - when our schedule messes up just because we mistake Sunday for Monday. Bid farewell to such a frustrating situation! This September calendar that starts on Monday will separate weekdays from weekends, making it easier to arrange your work during the week while sparing your precious Saturday and Sunday for relaxing.

This template promises to streamline your month since most, if not all, schools and businesses follow a standard calendar that begins on Monday. So, you can keep track of your schedule effortlessly and make the most out of every day. The calendar comes with a neat and simple design without distracting details, facilitating you to stay on top of every deadline. Your secret to a whole productive month is here!

September Holidays & Observances

When it comes to September, we all think of fruitful harvest days with nice weather. Indeed, you will have 30 days full of joy and peace when enjoying the autumnal vibe. There will also be other important events during the month, varying among regions and religions. But in general, we all celebrate the most important holiday - the Labor Day. We will offer you the September 2024 calendar with holidays to help you track down those events more easily:

Labor Day (Monday, September 2): A federal holiday in the United States and a state holiday across all states, Labor Day honors the contributions and achievements of American workers. It marks the unofficial end of summer, with many people enjoying parades, barbecues, and family gatherings.

Constitution Day and Citizenship Day (Tuesday, September 17): An observance celebrating the signing of the U.S. Constitution and recognizing those who have become U.S. citizens. It's a day to reflect on the rights and responsibilities of citizenship.

Astronomical Events in September 2024

The astronomical highlight of September 2024 includes the September Equinox, occurring around the 22nd or 23rd, marking the start of fall in the Northern Hemisphere and spring in the Southern Hemisphere. This event brings days and nights of almost equal length worldwide. Additionally, the Harvest Moon, September's Full Moon, provides a bright backdrop for the early fall nights, traditionally coinciding with the time to harvest crops.
Phases of the Moon:

New Moon (Monday, September 2): The New Moon marks the beginning of the lunar cycle, a time for setting new intentions and starting fresh projects.

1st Quarter (Wednesday, September 11): The First Quarter Moon is a great time for taking action on the plans and intentions set during the New Moon.

Full Moon (Tuesday, September 17): The Full Moon brings the lunar cycle to its peak, with the entire face of the Moon illuminated, often associated with completion and celebration.

3rd Quarter (Tuesday, September 24): The Third Quarter Moon signals the beginning of the end of the lunar cycle, a time for reflection and letting go.

Notable Events in September

September has been a month of significant historical and cultural milestones, shaping the world as we know it today. Here are some key events tied to specific dates:

September 2, 1945: On this day, World War II officially ended when Japan formally surrendered to the Allies, marking the end of one of the most devastating conflicts in human history. The surrender ceremony took place aboard the USS Missouri in Tokyo Bay.

September 5, 1972: The Munich Massacre occurred during the Summer Olympics in Munich, Germany, when members of the Palestinian group Black September took 11 Israeli athletes hostage. The tragic event ended in the deaths of all the hostages and highlighted the vulnerability of global events to terrorism.

September 8, 1888: The first known victim of the infamous serial killer Jack the Ripper, Mary Ann Nichols, was found in London's Whitechapel district. This marked the beginning of a series of brutal murders that would go down in history as one of the most notorious unsolved cases.

September 8, 1966: The first episode of "Star Trek" aired on TV, marking the beginning of a cultural phenomenon.

September 9, 1850: California was admitted as the 31st state of the United States. This event is commemorated each year as California Admission Day, celebrating the state's rich history and its pivotal role in the development of the nation.

September 11, 2001: The United States was rocked by the deadliest terrorist attacks in its history, as hijackers flew planes into the World Trade Center in New York City, the Pentagon in Arlington, Virginia, and a field in Pennsylvania. The events of 9/11 led to significant changes in global security policies and have had a lasting impact on the world.

September 17, 1787: The U.S. Constitution was signed by 39 delegates to the Constitutional Convention in Philadelphia, Pennsylvania. This document laid the foundation for the United States government and is celebrated annually as Constitution Day and Citizenship Day.

September 22, 1862: President Abraham Lincoln issued the preliminary Emancipation Proclamation, declaring that all slaves in Confederate-held territory would be free as of January 1, 1863. This was a crucial step toward the abolition of slavery in the United States.

September 27, 1821: Mexico gained independence from Spain after 11 years of war, marking the end of Spanish rule in the country. This date is celebrated as Mexican Independence Day, a key national holiday in Mexico.

These events highlight the historical and cultural significance of September, reminding us of the moments that have shaped our world. From the end of global conflicts to the signing of foundational documents, September is a month rich with history and reflection.

September Overview

September marks a transition from the warmth of summer to the crispness of autumn. The weather begins to cool, with temperatures gradually dropping, making it an ideal time for outdoor activities like hiking and enjoying the fall foliage. This month also signals the return to school for students across the country, bringing with it a renewed focus on routines and responsibilities. In addition to the back-to-school hustle, September often involves the start of football season, apple picking, and preparing for the upcoming holiday season. The days start to shorten, and the cozy atmosphere of early fall sets the tone for productivity and reflection as summer winds down.

History of September

September's name originates from the Latin word "Septem," meaning seven, reflecting its position in the ancient Roman calendar as the seventh month. The shift to the Gregorian calendar, initiated in 153 BCE when the new year moved to January, positioned September as the ninth month. This historical transition underscores the evolution of timekeeping and calendar organization from ancient Rome to present day.

September Birthstone and Birth Flowers

The sapphire, symbolizing wisdom, virtue, and good fortune, is September's birthstone. This gemstone is accompanied by September's birth flowers: the forget-me-not, morning glory, and aster. These flowers represent love, affection, and a strong sense of integrity, mirroring the deep blue of the sapphire.

Zodiac Signs in September

Virgo dominates the first part of September (until September 22), emphasizing traits of meticulousness, reliability, and practicality. Following Virgo, Libra begins on September 23, marking a period characterized by balance, harmony, and sociability. These signs reflect the transitional essence of September, bridging the gap between the seasons with a blend of contemplation and equilibrium.

Conclusion

Whether you are a worker with tons of workload or a student with an overflowing schedule, a September 2024 calendar will be a handy compass to guide you through the month. With user-friendly designs and customizable features, these templates offer a practical solution for planning and scheduling monthly activities. By harnessing their power, you can cultivate better time management habits, enhance productivity, and embark on a path toward greater success in both your personal and professional endeavors.

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Printable September 2024 Calendar Templates With Holidays

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Printable September 2024 Calendar Templates With Holidays. These free September Calendars are .pdf files that download and print on almost any printer.

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IXL | Compare animal life cycles | 5th grade science

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Improve your science knowledge with free questions in “Compare animal life cycles” and thousands of other science skills.

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Life cycles are taught in one grade level or another, whether students are in preschool hatching chicks or in middle school looking at the frog life cycle before dissecting one. Since students seem to constantly be learning about life cycles in one grade level or another. How can you make your lessons a little different? What life cycle activities can you use in your classroom to make life cycles a little more entertaining for students? Interactive ones are the best option!

6 Interactive Life Cycle Activities

When developing your life cycle lesson plans, you want to incorporate as many interactive activities as possible. Students are more likely to internalize the information when interacting and engaging in the lesson and/or activity. So whether you are working with preschoolers on the chicken life cycle or 8th graders and teaching all the life cycles, using life cycle activities that require student interaction are best.

Interactive Life Cycle Books

Reading a book to students about the life cycle of a butterfly life cycle or plant life cycle is helpful. But, having students interact with the book is even more effective. After each of the life cycle stages, give students a page where they must use their knowledge of the stage and fill in the blanks. You could use this page as a quick informal assessment or complete it as a group activity.

Some interactive life cycle books are:

Life Cycle Experiments

If learning about a ladybug life cycle or butterfly life cycle, consider bringing in a butterfly or ladybug rearing kit. Both can easily be found online and are simple to take care of. Students will love observing them. There are even frog hatchery kits if you're feeling brave! Try growing plants in light and complete darkness. How do their life cycles differ? Wheat seeds are great for this because they grow quickly. You can also grow bean plants inside a Ziploc baggie on paper towels. Students can easily see the steps of the plant life cycle through this experiment.

Anchor Charts

An anchor chart is a great way to get students involved in the life cycle unit. Once you have discussed each stage in a life cycle, create an anchor chart to hang in the classroom for students to review when they need it. Having students color, write, or draw the arrows on the anchor chart can make it interactive and fun. You could even put students into groups and have them each create their anchor to help them.

Life Cycle Games and Video

After getting into your life cycle unit, it's always fun to let students play games that help them remember each life cycle step. Ecosystem For Kids has over 30 different interactive life cycle games available. Plus, under each game, there are multiple paragraphs about the animal or insect the life cycle.

If you are looking for some great life cycle videos to add to your unit plan, check out some of these:

Life Cycle Project Ideas

If you are looking for some life cycle projects or life cycle craft ideas. There are a ton of different options. An easy project is to have students create life cycle folders. The folder can show all parts of the life cycle using the interactive books mentioned above.

Some other fun life cycle projects or crafts are:

Life Cycle Readings

If you are teaching older students, it is best to have some life cycle worksheets on their level. Using these close reading passages, students will do a close read about complete and incomplete metamorphosis. You then have the option to have students use a graphic organizer to show what they learned, answer multiple choice questions, or other. It's the perfect way to connect science and literacy. Plus, this is a high-interest, non-fiction passage and a great way to teach life cycles.

Life Cycle Activities

Teaching life cycles is a big task. There are so many of them, and finding ways to keep students engaged can take time and effort. But, using one of these 6 interactive life cycle activities will ensure students are not only listening but are learning and interacting with the materials. This will not only make the lessons more enjoyable for students but make teaching more fun for you too!

Looking for other science curriculum blog posts?
Teaching States of Matter Water Cycle Tips Natural Disaster Ideas

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6 Engaging and Interactive Life Cycle Activities For Kids

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Life cycles are taught in one grade level or another, whether students are in preschool hatching chicks or in middle school looking at the frog life cycle before dissecting one. Since students seem to constantly be learning about life cycles in one grade level or another.

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Quadratic equations are second-degree algebraic expressions and are of the form ax ² + bx + c = 0. The word " Quadratic" is derived from the word " Quad" which means square. In other words, a quadratic equation is an " equation of degree 2." There are many scenarios where a quadratic equation is used. Did you know that when a rocket is launched, its path is described by a quadratic equation? Further, a quadratic equation has numerous applications in physics, engineering, astronomy.

The quadratic equations are second-degree equations in x that have two answers for x. These two answers for x are also called the roots of the quadratic equations and are designated as (α, β). We shall learn more about the roots of a quadratic equation in the below content.

What is Quadratic Equation?

A quadratic equation is an algebraic expression of the second degree in x. The quadratic equation in its standard form is ax ² + bx + c = 0, where a, b are the coefficients, x is the variable, and c is the constant term. The first condition for an equation to be a quadratic equation is the coefficient of x ² is a non-zero term(a ≠0). For writing a quadratic equation in standard form, the x 2 term is written first, followed by the x term, and finally, the constant term is written. The numeric values of a, b, c are generally not written as fractions or decimals but are written as integral values.

Further in real math problems the quadratic equations are presented in different forms: (x - 1)(x + 2) = 0, -x ² = -3x + 1, 5x(x + 3) = 12x, x ³ = x(x ² + x - 3). All of these equations need to be transformed into standard form of the quadratic equation before performing further operations.

Quadratic Equation Formula

Quadratic Formula is the simplest way to find the roots of a quadratic equation. There are certain quadratic equations that cannot be easily factorized, and here we can conveniently use this quadratic formula to find the roots in the quickest possible way. The roots of the quadratic equation further help to find the sum of the roots and the product of the roots of the quadratic equation. The two roots in the quadratic formula are presented as a single expression. The positive sign and the negative sign can be alternatively used to obtain the two distinct roots of the equation.

Quadratic Formula = [-b ± √(b² - 4ac)]/2a

Important Formulas for Solving Quadratic Equations

The following list of important formulas is helpful to solve quadratic equations.

The quadratic equation in its standard form is ax² + bx + c = 0

The discriminant of the quadratic equation is D = b2 - 4ac

For D > 0 the roots are real and distinct.

For D = 0 the roots are real and equal.

For D < 0 the roots do not exist, or the roots are imaginary.

The formula to find the roots of the quadratic equation is x = \( \frac{-b \pm\sqrt{b^2 - 4ac}}{2a}\).

The sum of the roots of a quadratic equation is α + β = -b/a = - Coefficient of x/ Coefficient of x².

The product of the Root of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x²

The quadratic equation having roots α, β, is x² - (α + β)x + αβ = 0.

The condition for the quadratic equations \(a_1x^2 + b_1x + c_1 = 0\), and \(a_2x^2 + b_2x + c_2 = 0\) having the same roots is \(\dfrac{(a_1b_2 - a_2b_1)}{(b_1c_2 - b_2c_1)}\) = \((a_2c_1 - a_1c_2)^2\).

For positive values of a (a > 0), the quadratic expression f(x) = ax2 + bx + c has a minimum value at x = -b/2a.

For negative value of a (a < 0), the quadratic expression f(x) = ax2 + bx + c has a maximum value at x = -b/2a.

For a > 0, the range of the quadratic equation ax² + bx + c = 0 is [b² - 4ac/4a, ∞)

For a < 0, the range of the quadratic equation ax² + bx + c = 0 is : (∞, -(b² - 4ac)/4a]

Quadratic Formula Proof

Consider an arbitrary quadratic equation: ax ² + bx + c = 0, a ≠ 0

To determine the roots of this equation, we proceed as follows:

Now, we express the left hand side as a perfect square, by introducing a new term (b/2a) ² on both sides:

The left hand side is now a perfect square:

This is good for us, because now we can take square roots to obtain:

x + b/2a = ±√(b ² - 4ac)/2a

x = (-b ± √(b ² - 4ac))/2a

Thus, by completing the squares, we were able to isolate x and obtain the two roots of the equation.

Roots of a Quadratic Equation

The roots of a quadratic equation are the two values of x, which are obtained by solving the quadratic equation. The roots of a quadratic equation are referred to by the symbols alpha (α), and beta (β). These roots of the quadratic equation are also called the zeros of the equation. Here we shall learn more about how to find the nature of roots of a quadratic equation without actually finding the roots of the equation. And also check out the formulas to find the sum and the product of the roots of the equation.

Nature of Roots of the Quadratic Equation

The nature of roots of a quadratic equation can be found without actually finding the roots (α, β) of the equation. This is possible by taking the discriminant value, which is part of the formula to solve the quadratic equation. The value b ² - 4ac is called the discriminant of a quadratic equation, and is designated as 'D'. Based on the discriminant value the nature of the roots of the quadratic equation can be predicted.

Discriminant: D = b² - 4ac

D > 0, the roots are real and distinct

D = 0, the roots are real and equal.

D < 0, the roots do not exist or the roots are imaginary.

Relationship Between Coefficients and Roots of Quadratic Equation

The coefficient of x ², x term, and the constant term of the quadratic equation ax ²+ bx + c = 0 are useful to study more about the properties of roots of the quadratic equation. The sum and product of roots of a quadratic equation can be directly calculated from the equation, without actually finding the roots of the quadratic equation. The sum of the roots of the quadratic equation is equal to the negative of the coefficient of x divided by the coefficient of x ². The product of the root of the equation is equal to the constant term divided by the coefficient of the x ². For a quadratic equation ax ² + bx + c = 0, the sum and product of the roots are as follows.

Sum of the Roots: α + β = -b/a = - Coefficient of x/ Coefficient of x²

Product of the Roots: αβ = c/a = Constant term/ Coefficient of x²

The quadratic equation can also be formed for the given roots of the equation. If α, β, are the roots of the quadratic equation, then the quadratic equation is as follows.
x² - (α + β)x + αβ = 0
Methods to Solve Quadratic Equations

A quadratic equation can be solved to obtain two values of x or the two roots of the equation. There are four different methods to find the roots of the quadratic equation. The four methods of solving the quadratic equations are as follows.

Factorizing of Quadratic Equation

Formula Method of Finding Roots

Method of Completing the Square

Graphing Method to Find the Roots

Let us look in detail at each of the above methods to understand how to use these methods, their applications, and their uses.

Factorization of Quadratic Equation

Factorization of quadratic equation follows a sequence of steps. For a general form of the quadratic equation ax ² + bx + c = 0, we need to first split the middle term into two terms, such that the product of the terms is equal to the constant term. Further, we can take the common terms from the available term, to finally obtain the required factors. For understanding factorization, the general form of the quadratic equation can be presented as follows.

Let us understand factorization through the below example.

Thus the two obtained factors of the quadratic equation are (x + 2) and (x + 3).

Quadratic Formula to Find Roots

The quadratic equations which cannot be solved through the method of factorization can be solved with the help of a formula. The formula to solve the quadratic equation uses the terms from the standard form of a quadratic equation. Through the below formula we can obtain the two roots of x by first using the positive sign in the formula and then using the negative sign. Any quadratic equation can be solved using this formula.

Further to the above-mentioned two methods of solving quadratic equations, there is another important method of solving a quadratic equation. The method of completing the square for a quadratic equation is also useful to find the roots of the equation. This method includes numerous algebraic calculations and hence has been explained as a separate topic.

Method of Completing the Square

The method of completing the square for a quadratic equation, is to algebraically square and simplify, to obtain the required roots of the equation. Consider a quadratic equation ax ² + bx + c = 0, a ≠ 0. To determine the roots of this equation, we simplify it as follows:

Now, we express the left hand side as a perfect square, by introducing a new term (b/2a) ² on both sides:

Now with this method of completing the square, we could consolidate the value for the roots of the equation. Further on simplification and taking the square root, the two possible roots of the quadratic equation are, x = (-b + √(b ²- 4ac))/2a. Here the '+' sign gives one root and the '-' sign gives another root of the quadratic equation. Generally, this detailed method is avoided, and only the formula is used to obtain the required roots.

Graphing a Quadratic Equation

The graph of the quadratic equation ax ² + bx + c = 0 can be obtained by representing the quadratic equation as a function y = ax ² + bx + c. Further on solving and substituting values for x, we can obtain values of y, we can obtain numerous points. These points can be presented in the coordinate axis to obtain a parabola-shaped graph for the quadratic equation.

The point where the graph cuts the horizontal x-axis is the solution of the quadratic equation. These points can also be algebraically obtained by equalizing the y value to 0 in the function y = ax ² + bx + c and solving for x.

Quadratic Equations Having Common Roots

The two quadratic equation having common roots are \(a_1x^2 + b_1x + c_1 = 0\), and \(a_2x^2 + b_2x + c_2 = 0\). Let us solve these two equations to find the conditions for which these equations have a common root. The two equations are solved for x ² and x respectively.

\(\dfrac{x^2}{(b_1c_2 - b_2c_1)}\) = \(\dfrac{-x}{(a_1c_2 - a_2c_1)}\) = \(\dfrac{1}{(a_1b_2 - a_2b_1)}\)

\(x^2 =\dfrac{(b_1c_2 - b_2c_1)}{(a_1b_2 - a_2b_1)}\)

\(x=\dfrac{(a_2c_1 - a_1c_2)}{(a_1b_2 - a_2b_1)}\)

Hence of simplifying the above two expressions we have the following condition for the the two equations having the common root.

\(\dfrac{(a_1b_2 - a_2b_1)}{(b_1c_2 - b_2c_1)}\) = \((a_2c_1 - a_1c_2)^2\)

Maximum and Minimum Value of Quadratic Expression

The maximum and minimum value for the quadratic equation ax ² + bx + c = 0 can be observed in the below graphs. For positive values of a (a > 0), the quadratic expression has a minimum value at x = -b/2a, and for negative value of a (a < 0), the quadratic expression has a maximum value at x = -b/2a.

The maximum and minimum values of the quadratic expressions are of further help to find the range of the quadratic expression: The range of the quadratic expressions also depends on the value of a. For positive values of a( a > 0), the range is [ f(-b/2a), ∞), and for negative values of a ( a < 0), the range is (-∞, f(-b/2a)].

Solving a Quadratic Equation - Tips and Tricks

Some of the below-given tips and tricks on quadratic equations are helpful to more easily solve quadratic equations.

1. The quadratic equations are generally solved through factorization. But in instances when it cannot be solved by factorization, the formula is used.

2. The roots of a quadratic equation are also called the zeroes of the equation.

3. For quadratic equations having negative discriminant values, the roots are represented with the help of complex numbers.

4. The sum and product of roots of a quadratic equation can be used to find higher algebraic expressions involving these roots.

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FAQs on Quadratic Equation

What is a Quadratic Equation?

A quadratic equation in math is a second-degree equation of the form ax² + bx + c = 0. Here a, b, are the coefficients, c is the constant term, and x is the variable. Since the variable x is of the second degree, there are two roots or answer for this quadratic equation. The roots of the quadratic equation can be found by either solving by factorizing or through the use of a formula.

What is the Quadratic Formula?

The quadratic formula to solve a quadratic equation ax² + bx + c = 0 is x = [-b ± √(b² - 4ac)]/2a. Here we obtain the two values of x, by applying the plus and minus symbol in this formula. Hence the two possible values of x are [-b + √(b² - 4ac)]/2a, and [-b - √(b² - 4ac)]/2a.

How to Apply Quadratic Formula?

The values of a, b, and c are applied to the quadratic formula x = [-b ± √(b² - 4ac)]/2a, to obtain the two roots of the quadratic equation.

What is Determinant in Quadratic Formula?

The value b² - 4ac is called the determinant and is designated as D. The determinant is part of the quadratic formula. The determinants help us to find the nature of the roots of the quadratic equation, without actually finding the roots of the quadratic equation.

What are Some Real Life Applications of Quadratic Equations

Quadratic equations are used to find the zeroes of the parabola and its axis of symmetry. There are many real-world applications of quadratic equations. For instance, it can be used in running time problems to evaluate the speed, distance or time while traveling by car, train or plane. Quadratic equations describe the relationship between quantity and the price of a commodity. Similarly, demand and cost calculations are also considered quadratic equation problems. It can also be noted that a satellite dish or a reflecting telescope has a shape that is defined by a quadratic equation.

How are Quadratic Equations Different From Linear Equations?

A linear degree is an equation of a single degree and one variable, and a quadratic equation is an equation in two degrees and a single variable. A linear equation is of the the form ax + b = 0 and a quadratic equation is of the form ax² + bx + c = 0. A linear equation has a single root and a quadratic equation has two roots or two answers. Also, a quadratic equation is a product of two linear equations.

How to Simplify a Quadratic Equation?

The first step in the process of simplifying a quadratic equation is to transform it into the standard form ax² + bx + c = 0. Further, it can be simplified by finding its factors through the process of factorization. Also for an equation for which it is difficult to factorize, it is solved by using the formula. Additionally, there are a few other ways of simplifying a quadratic equation.

What Are the 4 Ways To Solve A Quadratic Equation?

The four ways of solving a quadratic equation are as follows.

Factorizing method

Formula Method

Method of Completing Squares

Graphing Method

How Do you Solve a Quadratic Equation By Factoring?

The quadratic equation can be solved by factorization through a sequence of three steps. First split the middle term, such that the product of the split terms is equal to the product of the first and the last terms. Let us assume the quadratic equation is of the form x² + (a + b)x + ab = 0, and it can be split as x² + ax + bx + ab = 0. As a second step, take the common term from the first two and the last two terms. x(x + a) + b(x + a) = 0, (x + a)(x + b) = 0. Finally equalize each of the factors to zero and obtain the x values. x + a = 0 and x + b = 0, and hence we can obtain x = -a and x = -b

How to Solve a Quadratic Equation by Completing the Square?

The quadratic equation solved by the method of completing the square and it uses the formula (a + b)² = a²

How to Find the Value of the Discriminant?

The value of the discriminant in a quadratic equation can be found from the variables and constant terms of the standard form of the quadratic equation ax² + bx + c = 0. The value of the discriminant is D = b² - 4ac, and it helps to predict the nature of roots of the quadratic equation, without actually finding the roots of the equation.

How Do You Solve Quadratic Equations With Graphing?

The quadratic equation can be solved similarly to a linear equal by graphing. Let us take the quadratic equation ax² + bx + c = 0 as y = ax² + bx + c . Here we take the set of values of x and y and plot the graph. These two points where this graph meets the x-axis, are the possible solutions of this quadratic equation.

How To Find Nature of Roots of Quadratic Equation?

The discriminant is helpful to predict the nature of the roots of the quadratic equation. The determinant of a quadratic equation ax² + bx + c = 0 is b² - 4ac. The discriminant is referred as D = b² - 4ac. If D > 0 the roots are real and distinct, for D = 0 the roots are equal, and for D < 0 the roots are imaginary complex numbers.

How Important Is the Discriminant in Determining the Nature of Roots of Quadratic Equation?

The discriminant is very much needed to easily find the nature of roots of the quadratic equation. Without the discriminant, finding the nature of the roots of the equation is a long process, as we first need to solve the equation to find both the roots. Hence the discriminant is an important and the needed quantity, which helps to easily find the nature of roots of the quadratic equation.

When Do Quadratic Equations Have Equal Roots?

The given quadratic equation has equal roots if the discriminant is equal to zero. For a quadratic equation of the form ax² + bx + c = 0 the discriminant is D = b² - 4ac = 0. Here both are roots are equal and each has a value of x = -b/2a.

What is the Use of Determinants in Quadratic Formula?

The determinant (D = b² - 4ac) is useful to predict the nature of the roots of the quadratic equation. For D > 0, the roots are real and distinct, for D = 0 the roots are real and equal, and for D < 0, the roots do not exist or the roots are imaginary complex numbers. With the help of this determinant and with the least calculations, we can find the nature of the roots of the quadratic equation.

How Many Roots does Quadratic Equation Have?

It is a second-degree equation in x, and hence two roots are obtained. We can obtain these roots of a quadratic equation using the quadratic formula. One root can be obtained using the positive sign and we can get another root by applying the negative sign in the formula.

How do you Solve a Quadratic Equation without Using the Quadratic Formula?

There are two alternative methods to the quadratic formula. One method is to solve the quadratic equation through factorization, and another method is by completing the squares. In total there are three methods to find the roots of a quadratic equation.

How is Quadratic Formula used to Solve a Quadratic Equation?

The algebra formula (a + b)² = a² + 2ab + b² is used to solve the quadratic equation and derive the quadratic formula. This algebraic formula is used to manipulate the quadratic equation and derive the quadratic formula to find the roots of the equation.

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Quadratic Equation - Formula, Examples | Quadratic Formula

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Quadratic equations are second-degree algebraic expressions and are of the form ax 2 + bx + c = 0. The word " Quadratic" is derived from the word " Quad" which means square. In other words, a quadratic equation is an " equation of degree 2."

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