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Method

1. Put into form ax² + bx + c

2. a x c = ?

3. Find factors of ?

4. Choose factors that add/subtract to make b

5. Rewrite equations with factors as b

6. Factorise a + 1/2b and 1/2b + c

7. Write repeated brackets and outer terms in brackets

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Method

1. Rearrange the quadratic into format ax² + bx + c = 0

2. Use the formula :

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Method

1. Express in the form a(x+b)2 +c  (complete the square)

2. Rearrange to find x

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Method

1. Rearrange into ax² + bx + c

2. Use formula :

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Method

1. Use equation : b2 - 4ac

2. Decide which of the 3 states it fits with thereofre determining the number of real roots

3 States
b2 - 4ac > 0 Two real roots
b2 - 4ac = 0  One real root
b2 - 4ac < 0 No real roots

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Method
Express in form a(x+b)2 +c  (complete the square)
C= minimum/maximum value
b x -1 = x when it occurs

Key Information
-x2 graphs have maximum values
x2 graphs have minimum values

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Method - Linear Graph

1. Substitute x values until you find the trend

2. Plot and join with a straight line.

Method - Quadratic graphs

1. Factorise function and set each bracket to 0

2. solve to find 2 values of x (where the graph crosses the x axis)

3. Determine the graph shape using x2 coefficient.

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Method

1. Rearrange into ax² + bx + c

2. Factorise the expression

3. Sketch the function

4. Decide which part of the graph contains the solution

5. Write the inequality solution

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Combing inequalities with Nature of Roots

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Combing inequalities with Nature of Roots

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Method

1. Find a= b= c=

2. Put into equation b2 - 4ac

3. Expand brackets

4. Put into form ax² + bx + c

5. Factorise

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Method- Using Substitution

1. Rearrange one of the equations so that it equals x or y

2. Substitute changed equation into other

3. Simplify and solve

4. Substitute the value into one of the equations to find the other

Method - Finding Intersections

1. Rearrange one of the equations so that is equals x or y

2. Substitute changed equation into other

3. Simplify and put into form ax² + bx + c

4. Factorise

5. Substitute the values into one of the equations to find the other

Method - Determining the number of solutions
Rearrange one of the equations so that it equals x or y
Substitue changed equation into other
Simplify and put into form ax² + bx + c = 0
Use b2 - 4ac

Key Information
b2 - 4ac > 0 Lines intersect at two points
b2 - 4ac = Lines intersect at one point
b2 - 4ac < 0 Lines do not intersect

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Label Co-ordinates (X₁ , Y₁) (X₂ , Y₂)

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Method -  Equations of lines

1. Find gradient

2. Use equation: y-y₁ = m(x-x₁)

3. Rearrange to get y=mx+c

Method - Intersections

1. Solve simultaneous equation

2. Substitute value into one equation to find the other

Key Information
Questions may be worded like ' lines l₁ and l₂ intersect at point C. Show C has coordinates (x,y)
Method  - Axis Intersection

1. Substitute 0 into the equation at appropriate place

2. Simplify and solve

Key Information
When the line crosses they y axis, x=0
When the line crosses the x axis, y=0
Method - Angles between two lines

1. Plot co-ordinates and form a right angled triangle

2. Calculate the lengths of the lines needed/available

3. Use SOHCAHTOA to find necessary angle

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Method- Equation of a circle
Form 1: (x-a)2 + (x-b)2 =r²
Centre = (a,b)
Radius= √r2
Form 2 : x² + y² +2gx + 2fy +c =0
Centre = (-g, -f) (divide 2gx +2fx by 2 and swap signs)
Radius = √g² +f² - c
Method - Points on a circle

1. Substitute the coordinates of the given into into the circle equation

2. Determine which scenario it lies in

Key information
If the solution is negative, the point lies inside the circle.
If the solution is equal to 0, the point lies on the circle.
If the solution is positive, the point lies outside the circle.
Method - Circle and line intersect

1. Rearrange on equation

2. Put changed equation into other

3. Simplify and put into form ax² + bx + c

4. Factorise to find values

Key information
When they intersect twice, there will  be two sets of coordinates
Where there is one point of intersection ( a tangent), there is one set of coordinates
Method - No points of intersection

1. Rearrange one equation

2. Put changed equation into other

3. Simplify and put into form ax² + bx + c = 0

4. Use b2 - 4ac to prove that there are no real roots and therefore do not intersect

Method - Do Circles meet?
Find distance between centres (length of line)
Find radius of both circles
Key Information
No points of intersection = C₁C₂ > r₁ + r₂
Two points of intersection =C₁C2 < r₁ + r₂
Circles tough externally =C₁C2 = r₁ + r₂
Circles touch internally= C₁C₂ = r₁ - r₂
Orthogonal Circles =d² = r₁ 2 +  r₂ ²
(If the tangents are perpendicular at point of intersection they are said to be orthogonal)

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Method - First Principles

1. Use Fomula

2. Replace each x with (x+h)

3. f(x) = original value

4. Rearrange to remove fraction

5. Imagine h=0, simplify and solve

Method - The Quick Way

1. Bring the power down

2. Subtract 1 from the power

Eg function= y=axn
Derivative = n x axn-1
Method- Finding the tangent to a curve

1. Differentiate the function

2. Substitute the x co-ordinate into derivative. (This gives us the gradient of the tangent)

3. Use y-y₁=m(x-x₁) to find the equation of the tangent

Method - Finding the normal to a curve

1. Differentiate the function

2. Substitute x co-ordinate into derivative to find the gradient of the tangent

3. Find the gradient of the normal (-1 / gradient of tangent)

4. Use y-y₁=m(x-x₁) to find the equation of the normal

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Method- Finding stationary points

1. Differentiate the function

2. Set the derivative equal to 0 and solve the equation (usually factorise)

3. Substitute values into original equation to give two sets of co-ordinates

Key Information

A stationary point is a part of the graph where the first derivative is 0

Maximum - Represents the highest value of the function - change from increasing to decreasing

Minimum- Represents the lowest value of the function - change from decreasing to increasing

Inflection - No representation of a value - direction of function doesn’t change

Method- Classifying the stationary points

1. Differentiate the function

2. Set the derivativeequal to 0 and solve the equation (usually factorising)

3. Substitute values into original equation to give two sets of co-ordinates (the stationary points)

4. Find second derivative

5. Substitute x value into second derivative

Key Information

If d²y / dx² < 0 = maximum point

If d²y / dx² > 0 = minimum point

If d²y / dx2 = 0 = Can’t be classified - consider values either side of stationary points

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Method - Indefinite interval

1. Add 1 to index (power)

2. Divide by new power

Method - Definite interval

1. Add 1 to index (power

2. Divide by new power

3. Substitute in upper limit to integral

4. Substitute in lower limit

5. Upper limit answer - lower limit answer

Key Information

Indefinite interval - no numbers on integral - have to include a constant of integration (+c)

Definite interval - has number on integral called limits - when we substitute in limits we get a definitive solution

When integrating we start with f(x).

After integration we have F(x)

Method - Area under a curve

1. Use edges of function (area required in graph) as limits

2. Integrate function

3. Substitute in limits

4. Upper limit answer - lower limit answer

Method - Area between a curve and a line
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Method

1. Rearrange into form sin θ

2. Find the angle

3. Choose quadrants where solutions exist, draw them in

4. Measure, anticlockwise, the angle from the x axis to each line

5. These are the solutions

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Useful when we have an angle and an opposite side and are given another side/angle

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Useful when we have all 3 sides and are finding a missing angle OR when we have 2 sides and an enclosed angle and are aiming to find the third side.

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1/2 absinC

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Roots of an equation

The root of an equation is where the curve crosses the x axis

Factor Theorem

A polynomial f(x) has factor (x-a) if and not if f(a)=0

If (x-a) is a factor, x=a is a root of the equation f(x)

Remainder Theorem

When a polynomial f(x) is divided by (ax-b) the remainder is f(b/a)

You replace x in your original equation with b/a and then calculate the appropriate value.

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Method

1. Divide highest power of x by highest power of x in divisor

2. Multiply back ‘new’ element in top by divisor and write underneath

3. Subtract two lines

4. Repeat process until you are left with a single number