Completing the square review

Completing the square is a technique for factoring quadratics. This article reviews the technique with examples and even lets you practice the technique yourself.

What is completing the square?

Completing the square is a technique for rewriting quadratics in the form (x+a)2+b(x+a)^2+b.
For example, x2+2x+3x^2+2x+3 can be rewritten as (x+1)2+2(x+1)^2+2. The two expressions are totally equivalent, but the second one is nicer to work with in some situations.

Example 1

We're given a quadratic and asked to complete the square.
x2+10x+24=0x^{2}+10x+24 = 0
We begin by moving the constant term to the right side of the equation.
x2+10x=24x^2 + 10x = -24
We complete the square by taking half of the coefficient of our xx term, squaring it, and adding it to both sides of the equation. Since the coefficient of our xx term is 1010, half of it would be 55, and squaring it gives us 25\blueD{25}.
x2+10x+25=24+25x^2 + 10x \blueD{ + 25} = -24 \blueD{ + 25}
We can now rewrite the left side of the equation as a squared term.
(x+5)2=1( x + 5 )^2 = 1
Take the square root of both sides.
x+5=±1x + 5 = \pm1
Isolate xx to find the solution(s).
x=5±1x = -5\pm1
Want to learn more about completing the square? Check out this video.

Example 2

We're given a quadratic and asked to complete the square.
4x2+20x+25=04x^{2}+20x+25 = 0
First, divide the polynomial by 44 (the coefficient of the x2x^2 term).
x2+5x+254=0x^2 + 5x + \dfrac{25}{4} = 0
Note that the left side of the equation is already a perfect square trinomial. The coefficient of our xx term is 55, half of it is 52\dfrac{5}{2}, and squaring it gives us 254\blueD{\dfrac{25}{4}}, our constant term.
Thus, we can rewrite the left side of the equation as a squared term.
(x+52)2=0( x + \dfrac{5}{2} )^2 = 0
Take the square root of both sides.
x+52=0x + \dfrac{5}{2} = 0
Isolate xx to find the solution.
The solution is: x=52x = -\dfrac{5}{2}

Practice

Problem 1
Complete the square to rewrite this expression in the form (x+a)2+b(x + a)^2 + b.
x22x+17x^{2}-2x+17

Want more practice? Check out these exercises:
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