Solve A Quadratic Equation by Completing the Square

What you will learn in this mini lesson

You will learn how to solve Quadratic Equations in Standard Form by Completing the Square. Our Step by Step Calculator allows you to Complete the Square and Solve your own quadratic equation.

Quick Example
2x2 -12x+16 Factor out 2:
= 2(x2 -6x+8)
= 2(x-3)2 -2 since 2(x-3)2 = 2x2-12x+18
To find Zeros: 2(x-3)2 = 2
(x-3)2 = 1 Take Square Root
(x-3) = ±1 Solve now
x=4 and x=2 . That’s all 😉




Solve A Quadratic Equation by Completing the Square

The Quadratic Equation in Standard Form is
 y=x²+bx+c 
To Solve by Completing the Square we add and subtract
(b/2)² which yields:
 y=x²+bx+ (b/2)² + c - (b/2)²
Completing the Square yields:
 y=(x+b/2)² + c - (b/2)²


Example: Solve A Quadratic Equation by Completing the Square

We are given the Quadratic Equation below in Standard Form
y=x²- 6x-7 .

First, add and subtract
(- b/( 2a)²= (-(-6)/(2*1))² = 3²=9 .

Thus, we have:
y=(x²- 6x+ 9)-7- 9

This allows us to Solve via Completing the Square:
y=(x-3)²-16 .

To solve (x-3)²-16 =0 we first add 16:

(x-3)²=16 . Next, we take the Square Root:

x-3 = ±4 . Adding 3 to ±4 yields the 2 solutions:

x=7 , x=-1 .

Easy, wasn’t it?

Tip: When using the Complete the Square Solver to solve
x²-6x-7=0 we must enter the 3 coefficients as
a=1, b=-6and c=-7.

Then, the Solver will first Complete the Square to find the Vertex
(h,k)=(3,-16) .

Thus, the Vertex Form of the Parabola is y=(x-3)²-16 .

Solving (x-3)²-16=0 yields the two zeros: x=7 , x=-1 .

Get it now? Try the above Complete the Square Solver again or check out it this excellent Step by Step Complete the Square video:


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