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“Sample of Solving Quadratic Equations by Completing the Square”

Complete the Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation (y−7) ²=12 because the left side was a perfect square.

(y−7) ²=12

y−7= ± √12

y−7= ± 2√3

y=7±2√3

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form (x−k) ² (x−k)2 in order to use the Square Root Property

What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?

Let’s look at this example to help us recognize the patterns.

(x+9) ²

(x+9)(x+9)

x²+9x+9x+81

x²+18x+81

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Solving Quadratic Equation By Completing the Squares

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Solving Quadratic Equation By Completing the Squares

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in completing the squares we must be familiar of the square of a binomial and perfect square trinomial.
we must remember that we only apply  completing the squares if the equation or the expression cannot be solved through extracting the square root or factoring techniques.
for example:
we have x ² + 6x + 1 = 0 , so this equation cannot be solved by factoring, because 1 doesn't have a factor that when we add it it will be equal to  6, we also can't use extracting the square root in this situation, therefor, we must solve this by completing the square,
1. first is we must isolate the constant,  transpose 1
it will be x ² + 6x = -1
2. we must make those a perfect square trinomial,
so x ² + 6x + _ = -1 + _ , to make this a PST we must divide the 6x into 2 , which is 3 squared by two is equal to 9
so  x ² + 6x + 9 = -1 + 9 now that this is a perfect square trinomial
3.  we must transform it into binomial
x ² + 6x + 9 = -1 + 9
we get the square root of x ² + 6x + 9   then -1 + 9 = 8  so we got  [X + 3]² = 8
4. now we remove the  square, by  using the extracting the square root method,
√[X + 3]² =√8
x + 3 = +-√8
5. now we simplify it to  x + 3 = +-4√2 SIMPLIFY AGAIN
x + 3 = +-2√2
then transpose 3 ,
X = +-2√2 -3 so this is our final answer.

BY GAURAKI

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Example of equation using the Zero Product Property

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Example of equation using the Zero Product Property

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1. x ²+8x=20
Transpose: x²+8x-20=0
Factor: (x-10) (x+2) - you can use the triangle method to find the factors
Solutions of the equation: 10, -2

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Example of equation using the Square Root Property

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Example of equation using the Square Root Property

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1. x²=169
Find the square root of: √x²=√169
x=13
The solutions of the equation: 13, -13

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The process in solving x² = d is referred to as taking the square root of the equation
If x² = d, then x= +√d

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A quadratic equation is an equation of the form  ax ² + bx + c = 0, where a, b, and c represent real numbers and a ≠ 0. In standard form a must be a positive real number. The solutions to a quadratic equation are called its roots.

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Zero Factor theorem:
- If p and q are algebraic expressions, then pq = 0, if and only if p=0 or q=0

This theorem states that  ax ² + bx + c = 0 can be written as a product of two-first degree polynomials, then the solutions can be found by setting each factor equal to zero.

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Solving Quadratic Equations through the Square Root Property

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Solving Quadratic Equations through the Square Root Property

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{ If x²=d, then x=±√d }
1.) Transpose if needed in order to isolate x²
2.) Simplify your equation
3.) Find the square root of both sides of your equation
4.) Consider two roots of the equation.

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Steps on Solving Quadratic Equations by the Zero Product Property 

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Steps on Solving Quadratic Equations by the Zero Product Property 

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Factoring Method:
these are the steps in solving quadratic equations by factoring:
1.) Write the equation in the form ax ² + bx + c = 0
2.) Factor the left-hand side of the equation.
3.) Apply the zero product theorem (set each factor equal to zero).
4.) Solve the equations
5.) Check the results in the original equation.