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# 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 $\left(x+a{\right)}^{2}+b$.
For example, ${x}^{2}+2x+3$ can be rewritten as $\left(x+1{\right)}^{2}+2$. The two expressions are totally equivalent, but the second one is nicer to work with in some situations.

### Example 1

${x}^{2}+10x+24=0$
We begin by moving the constant term to the right side of the equation.
${x}^{2}+10x=-24$
We complete the square by taking half of the coefficient of our $x$ term, squaring it, and adding it to both sides of the equation. Since the coefficient of our $x$ term is $10$, half of it would be $5$, and squaring it gives us $25$.
${x}^{2}+10x+25=-24+25$
We can now rewrite the left side of the equation as a squared term.
$\left(x+5{\right)}^{2}=1$
Take the square root of both sides.
$x+5=±1$
Isolate $x$ to find the solution(s).
$x=-5±1$

### Example 2

$4{x}^{2}+20x+25=0$
First, divide the polynomial by $4$ (the coefficient of the ${x}^{2}$ term).
${x}^{2}+5x+\frac{25}{4}=0$
Note that the left side of the equation is already a perfect square trinomial. The coefficient of our $x$ term is $5$, half of it is $\frac{5}{2}$, and squaring it gives us $\frac{25}{4}$, our constant term.
Thus, we can rewrite the left side of the equation as a squared term.
$\left(x+\frac{5}{2}{\right)}^{2}=0$
Take the square root of both sides.
$x+\frac{5}{2}=0$
Isolate $x$ to find the solution.
The solution is: $x=-\frac{5}{2}$

## Practice

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

Want more practice? Check out these exercises:

## Want to join the conversation?

• I find it frusterating and a bit unfair that in both the review and the videos, the problems they show are equations like 4x^2 +20x + 24, while the problems we are given in the practices include functions: h(x) = x^2 +3x -18.
Can someone give me advice on dealing with the functions? I know how they work, but how can I do something to "both sides of the equation" when there is only one side to begin with?
• As an expression, you learn to complete the square. As an equation, one of the main purposes of completing the square is to find the roots, so it is where h(x)=0 OR to get an equation in vertex form so that graphing is made easier.
• I am not understanding how to complete squares when fractions are involved could someone tell me a better way of understanding because the tips or hints given aren't really helping me understand.
• I recommend keeping everything in fraction form so basically writing 7/2 instead of 3.5, it makes it simpler to square.
• I learned how to do it a different way. I learned to take the b term and do (b/2) to the second power. Then add that term to both sides and simplify from there. I find this way much easier
• need to complete the square, the problem is x^2 +10x+blank
• its completing the square .so x^2 + 10 x + 25 - 25 which is equal to( x +5)^2 -25
• In problem 2, the question have the same answer in both X1 and X2. However the problem doesn't automatically consider the answer when it is in X1 = 10, while X2 = 4
• Notice that the problem reads, "Give the solutions in ascending order."
• "Give the solutions in ascending order". Hahaha... that caught me out. Was almost bursting blood vessels in my brain trying to figure out where I'd gone wrong... then I saw "ascending order" and realised I'd got the maths right but hadn't submitted the answers in ascending order. READ THE QUESTION FULLY is the moral of this tale!
• how to solve x^2-2x-24=0?
• First off you have to factor the trinomial.
(x-6)(x+4)=0
I got to that by using the Diamon Method of factoring. But it can be done other ways. One of the other methods can be found here. https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-quadratics-2/v/factoring-trinomials-by-grouping-4

After I have found the factor I can find the zeros.
So what you do is you take each half of the equation and set it equal to zero.

x-6=0
x+4=0

After you do this you solve for x.

x=6
x=-4

So the zeros of the formula and the partial answer to your question is x=6 and x=-4.
I hope this helped you.