Solve A Quadratic Equation by Completing the Square

Solve A Quadratic Equation by Completing the Square

Here is how to Solve by Completing the Square.
The Quadratic Equation in Standard Form is
\boxed{ y=x^2+bx+c }

To Solve by Completing the Square we add and subtract
({b \over 2})^2 which yields:
\boxed{ y=x^2+bx+ ({b \over 2})^2 + c - ({b \over 2})^2 }

Completing the Square yields:
\boxed{ y=(x+b/2)^2 + c - ({b \over 2})^2 }

which is the Complete the Square Formula.

Sample Problem: Solve A Quadratic Equation by Completing the Square

We are given the Quadratic Equation below in Standard Form
y=x^2- 6x-7 .

First, add and subtract
({ - b \over 2a})^2= ({ -(-6)\over (2*1)})^2 = 3^2=9 .

Thus, we have:
y=(x^2- 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=\pm 4 . Adding 3 to \pm 4 yields the 2 solutions:

x=7 , x=-1 .

Easy, wasn’t it?

Tip: When using the Complete the Square Solver to solve
x^2-6x-7=0 we must enter the 3 coefficients as
a=1, b=-6 and 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: