# Completing the Square

## Create perfect square trinomials using the additive property of equality

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Completing the Square

What if for homework you were given a perfect square trinomial to factor, which you wrote in your notebook. However, you accidentally dropped your notebook in a mud puddle, and the last term in the trinomial became obscured. All you can currently see are the first two terms, \begin{align*}x^2 + 16x\end{align*}. How would you go about finding the third term? Once you found the third term, if you factored the trinomial and set it equal to 0, what would be the solution(s) to the equation? In this Concept, you'll learn how to complete the square to solve quadratic equations like this one.

### Try This

Multimedia Link: Visit the http://www.mathsisfun.com/algebra/completing-square.html - mathisfun webpage for more explanation on completing the square.

### Guidance

Completing the square is a method used to create a perfect square trinomial, as you learned in the previous Concept.

A perfect square trinomial has the form \begin{align*}a^2+2(ab)+b^2\end{align*}, which factors into \begin{align*}(a+b)^2\end{align*}.

#### Example A

Find the missing value to create a perfect square trinomial: \begin{align*}x^2+8x+?\end{align*}.

Solution:

The value of \begin{align*}a\end{align*} is \begin{align*}x\end{align*}. To find \begin{align*}b\end{align*}, use the definition of the middle term of the perfect square trinomial.

\begin{align*}&& 2(ab) &= 8x\\ a \ \text{is} \ x, && 2(xb) &= 8x\\ \text{Solve for} \ b: && \frac{2xb}{2x} &= \frac{8x}{2x} \rightarrow b=4\end{align*}

To complete the square you need the value of \begin{align*}b^2\end{align*}.

\begin{align*}b^2=4^2=16\end{align*}

The missing value is 16. The perfect square trinomial we are looking for is \begin{align*}x^2+8x+16\end{align*}.

To complete the square, the equation must be in the form: \begin{align*}y=x^2+\left(\frac{1}{2} b \right )x+b^2\end{align*}.

Looking at the above example, \begin{align*}\frac{1}{2}(8)=4\end{align*} and \begin{align*}4^2=16\end{align*}.

#### Example B

Find the missing value to complete the square of \begin{align*}x^2+22x+\end{align*}?. Then factor.

Solution:

Use the definition of the middle term to complete the square.

\begin{align*}\frac{1}{2} (b)=\frac{1}{2} (22)=11\end{align*}

Therefore, \begin{align*}11^2=121\end{align*} and the perfect square trinomial is \begin{align*}x^2+22x+121\end{align*}. Rewriting in its factored form, the equation becomes \begin{align*}(x+11)^2\end{align*}.

Solve Using Completing the Square

Once you have completed the square, you can solve using the method learned in the last Concept.

#### Example C

Solve \begin{align*}x^2+22x+121=0\end{align*}.

Solution: By completing the square and factoring, the equation becomes:

\begin{align*}&& (x+11)^2 &= 0\\ \text{Solve by taking the square root:} && x+11 &= \pm0\\ \text{Separate into two equations:} && x+11 &=0 \ or \ x+11=0\\ \text{Solve for} \ x: && x &= -11\end{align*}

#### Example D

Solve \begin{align*}k^2-6k+8=0\end{align*}.

Solutions:

Using the definition to complete the square, \begin{align*}\frac{1}{2}(b)=\frac{1}{2}(-6)=-3\end{align*}. Therefore, the last value of the perfect square trinomial is \begin{align*}(-3)^2=9\end{align*}. The equation given is:

\begin{align*}k^2-6k+8=0, \ and \ 8 \neq 9\end{align*}

In order to factor, complete the square by subtracting 8 and then adding 9 to each side:

\begin{align*}k^2-6k=-8\end{align*}

Remember to use the Addition Property of Equality.

\begin{align*}&& k^2-6k+9 &= -8+9\\ \text{Factor the left side.} && (k-3)^2 &= 1\\ \text{Solve using square roots.} && \sqrt{(k-3)^2} &= \pm \sqrt{1}\\ && k-3 &=1 \ or \ k-3=-1\\ && k &= 4 \ or \ k=2\end{align*}

### Guided Practice

Solve \begin{align*}x^2+10x+9=0\end{align*}.

Solutions:

First, find \begin{align*}b\end{align*}:

\begin{align*}\frac{1}{2}(b)=\frac{1}{2}(10)=5\end{align*}

Therefore, the last value of the perfect square trinomial is \begin{align*}5^2=25\end{align*}. The equation given is:

\begin{align*}x^2+10x+9=0, \ and \ 9 \neq 25\end{align*}

In order to factor, we must turn the left side into a perfect square trinomial.

Subtract 9:

\begin{align*}x^2+10x=-9\end{align*}

Complete the square: Remember to use the Addition Property of Equality.

\begin{align*}&& x^2+10x+25 &= -9+25\\ \text{Factor the left side.} && (x+5)^2 &= 16\\ \text{Solve using square roots.} && \sqrt{(x+5)^2} &= \pm \sqrt{16}\\ && x+5 &=4 \ or \ x+5=-4\\ && x &= -1 \ or \ x=-9\end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Solving Quadratic Equations by Completing the Square (14:06)

1. What does it mean to “complete the square”?
2. Describe the process used to solve a quadratic equation by completing the square.

Complete the square for each expression.

1. \begin{align*}x^2+5x\end{align*}
2. \begin{align*}x^2-2x\end{align*}
3. \begin{align*}x^2+3x\end{align*}
4. \begin{align*}x^2-4x\end{align*}
5. \begin{align*}3x^2+18x\end{align*}
6. \begin{align*}2x^2-22x\end{align*}
7. \begin{align*}8x^2-10x\end{align*}
8. \begin{align*}5x^2+12x\end{align*}

Solve each quadratic equation by completing the square.

1. \begin{align*}x^2-4x=5\end{align*}
2. \begin{align*}x^2-5x=10\end{align*}
3. \begin{align*}x^2+10x+15=0\end{align*}
4. \begin{align*}x^2+15x+20=0\end{align*}
5. \begin{align*}2x^2-18x=0\end{align*}
6. \begin{align*}4x^2+5x=-1\end{align*}
7. \begin{align*}10x^2-30x-8=0\end{align*}
8. \begin{align*}5x^2+15x-40=0\end{align*}

Mixed Review

1. A ball dropped from a height of four feet bounces 70% of its previous height. Write the first five terms of this sequence. How high will the ball reach on its \begin{align*}8^{th}\end{align*} bounce?
2. Rewrite in standard form: \begin{align*}y=\frac{2}{7} x-11\end{align*}.
3. Graph \begin{align*}y=5 \left( \frac{1}{2} \right)^x\end{align*}. Is this exponential growth or decay? What is the growth factor?
4. Solve for \begin{align*}r: |3r-4| \le 2\end{align*}.
5. Solve for \begin{align*}m:-2m+6=-8(5m+4)\end{align*}.
6. Factor \begin{align*}4a^2+36a-40\end{align*}.

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