Completing the square is the process of rewriting a quadratic expression from the expanded form $x^2+2bx+c$x2+2bx+c into the square form $\left(x+b\right)^2+c-b^2$(x+b)2+c−b2. The name comes from how the method was first performed by ancient Greek mathematicians who were actually looking for numbers to complete a physical square.
There are two ways to think about the completing the square method: visually (using a square), and algebraically. We will see that we can use this method to find the solutions to the quadratic equation and to more easily graph the function.
Consider the following quadratics:
$x^2+4x-5$x2+4x−5
$x^2+2x-3$x2+2x−3
$x^2-8x-20$x2−8x−20
Let's imagine we want to factorise them using the complete the square method.
The steps involved are:
As this is a visual method, its best to learn by watching the process in action, so here are three examples.
We wish to solve $x^2+4x-5=0$x2+4x−5=0 by first completing the square.
Completing the square on $x^2+4x-5$x2+4x−5:
Then we solve this algebraically.
$x^2+4x-5$x2+4x−5 | $=$= | $0$0 |
$\left(x+2\right)^2-9$(x+2)2−9 | $=$= | $0$0 |
$\left(x+2\right)^2$(x+2)2 | $=$= | $9$9 |
$x+2$x+2 | $=$= | $\pm3$±3 |
Then by subtracting $2$2 from both sides of the equation we have $x=1$x=1 or $x=-5$x=−5.
We wish to solve $x^2+2x-3=0$x2+2x−3=0 by first completing the square.
Completing the square on $x^2+2x-3$x2+2x−3:
Then we solve this algebraically.
$x^2+2x-3$x2+2x−3 | $=$= | $0$0 |
$\left(x+1\right)^2-4$(x+1)2−4 | $=$= | $0$0 |
$\left(x+1\right)^2$(x+1)2 | $=$= | $4$4 |
$x+1$x+1 | $=$= | $\pm2$±2 |
And by subtracting $1$1 from both sides of the equation we have $x=1$x=1 or $x=-3$x=−3.
We wish to solve $x^2-8x-20=0$x2−8x−20=0 by first completing the square.
Completing the square on $x^2-8x-20$x2−8x−20:
Then we solve this algebraically.
$x^2-8x-20$x2−8x−20 | $=$= | $0$0 |
$\left(x-4\right)^2-36$(x−4)2−36 | $=$= | $0$0 |
$\left(x-4\right)^2$(x−4)2 | $=$= | $36$36 |
$x-4$x−4 | $=$= | $\pm6$±6 |
We can add 4 to both sides of the equation to get $x=10$x=10 or $x=-2$x=−2.
This interactive will help you to visualise the process. Watch this video for an explanation .
Let's imagine we want to factorise the same quadratics as before, but this time using the algebraic version of the complete the square method.
The steps involved are:
Solve the following quadratic equation by completing the square:
$x^2+18x+32=0$x2+18x+32=0
Solve $x^2-6x-16=0$x2−6x−16=0 by completing the square:
Solve for $x$x by first completing the square.
$x^2-2x-32=0$x2−2x−32=0
Solve the following quadratic equation by completing the square:
$4x^2+11x+7=0$4x2+11x+7=0
Write all solutions on the same line, separated by commas.
Enter each line of work as an equation.