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# Completing the Square

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

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Practice Completing the Square
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Completing the Square When the Leading Coefficient Doesn't Equal 1

The area of another parallelogram is given by the equation 3x2+9x5=0\begin{align*}3x^2 + 9x - 5 = 0\end{align*}, where x is the length of the base. What is the length of this base?

### Guidance

When there is a number in front of x2\begin{align*}x^2\end{align*}, it will make completing the square a little more complicated. See how the steps change in Example A.

#### Example A

Determine the number c that completes the square of 2x28x+c\begin{align*}2x^2 - 8x + c \end{align*}.

Solution: In the previous concept, we just added (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}, but that was when a=1\begin{align*}a=1\end{align*}. Now that a1\begin{align*}a \neq 1\end{align*}, we have to take the value of a into consideration. Let's pull out the GCF of 2 and 8 first.

2(x24x)\begin{align*}2 \left(x^2 - 4x \right)\end{align*}

Now, there is no number in front of x2\begin{align*}x^2\end{align*}.

(b2)2=(42)2=4\begin{align*}\left(\frac{b}{2}\right)^2 = \left(\frac{4}{2}\right)^2 = 4\end{align*}.

Add this number inside the parenthesis and distribute the 2.

2(x24x+4)=2x24x+8\begin{align*}2 \left(x^2 - 4x +4 \right)=2x^2-4x+8\end{align*}

So, c=8\begin{align*}c=8\end{align*}.

#### Example B

Solve 3x29x+11=0\begin{align*}3x^2-9x+11=0\end{align*}

Solution:

1. Write the polynomial so that x2\begin{align*}x^2\end{align*} and x\begin{align*}x\end{align*} are on the left side of the equation and the constants on the right.

3x29x=11

2. Pull out a\begin{align*}a\end{align*} from everything on the left side. Even if b\begin{align*}b\end{align*} is not divisible by a\begin{align*}a\end{align*}, the coefficient of x2\begin{align*}x^2\end{align*} needs to be 1 in order to complete the square.

3(x23x+)=11

3. Now, complete the square. Determine what number would make a perfect square trinomial.

To do this, divide the x\begin{align*}x-\end{align*}term by 2 and square that number, or (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.

(b2)2=(32)2=94

4. Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add a(b2)2\begin{align*}{\color{red}a\cdot\left(\frac{b}{2}\right)^2}\end{align*} to keep the equation balanced.

3(x23x+94)=11+274

5. Factor the left side and simplify the right.

3(x32)2=174

6. Solve by using square roots.

(x32)2x32x=1712=±i172333=32±516i

Be careful with the addition of step 2 and the changes made to step 4. A very common mistake is to add (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*} to both sides, without multiplying by a\begin{align*}a\end{align*} for the right side.

#### Example C

Solve 4x2+7x18=0\begin{align*}4x^2+7x-18=0\end{align*}.

Solution: Let’s follow the steps from Example B.

1. Write the polynomial so that x2\begin{align*}x^2\end{align*} and x\begin{align*}x\end{align*} are on the left side of the equation and the constants on the right.

4x27x=18

2. Pull out a\begin{align*}a\end{align*} from everything on the left side.

4(x2+74x+)=18

3. Now, complete the square. Find (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.

(b2)2=(78)2=4964

4. Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add a(b2)2\begin{align*}{\color{red}a\cdot\left(\frac{b}{2}\right)^2}\end{align*} to keep the equation balanced.

4(x2+74x+4964)=18+4916

5. Factor the left side and simplify the right.

4(x+78)2=33716

6. Solve by using square roots.

(x+78)2x+78x=33764=±3378=78±3378

Intro Problem Revisit We can't factor \begin{align*}3x^2 + 9x - 5 = 0\end{align*}, so let's follow the step-by-step process we learned in this lesson.

1. Write the polynomial so that \begin{align*}x^2\end{align*} and \begin{align*}x\end{align*} are on the left side of the equation and the constants on the right.

2. Pull out \begin{align*}a\end{align*} from everything on the left side.

3. Now, complete the square. Find \begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.

4. Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add \begin{align*}{\color{red}a\cdot\left(\frac{b}{2}\right)^2}\end{align*} to keep the equation balanced.

5. Factor the left side and simplify the right.

6. Solve by using square roots.

However, because x is the length of the parallelogram's base, it must have a positive value. Only \begin{align*}x =-\frac{3}{2} + \frac{{\sqrt{141}}}{6}\end{align*} results in a positive value, so the length of the base is \begin{align*}x =-\frac{3}{2} + \frac{{\sqrt{141}}}{6}\end{align*}.

### Guided Practice

Solve the following quadratic equations by completing the square.

1. \begin{align*}5x^2+29x-6=0\end{align*}

2. \begin{align*}8x^2-32x+4=0\end{align*}

Use the steps from the examples above to solve for \begin{align*}x\end{align*}.

1.

2.

### Explore More

Solve the quadratic equations by completing the square.

1. \begin{align*}6x^2-12x-7=0\end{align*}
2. \begin{align*}-4x^2+24x-100=0\end{align*}
3. \begin{align*}5x^2-30x+55=0\end{align*}
4. \begin{align*}2x^2-x-6=0\end{align*}
5. \begin{align*}\frac{1}{2}x^2+7x+8=0\end{align*}
6. \begin{align*}-3x^2+4x+15=0\end{align*}

Solve the following equations by factoring, using square roots, or completing the square.

1. \begin{align*}4x^2-4x-8=0\end{align*}
2. \begin{align*}2x^2+9x+7=0\end{align*}
3. \begin{align*}-5(x+4)^2-19=26\end{align*}
4. \begin{align*}3x^2+30x-5=0\end{align*}
5. \begin{align*}9x^2-15x-6=0\end{align*}
6. \begin{align*}10x^2+40x+88=0\end{align*}

Problems 13-15 build off of each other.

1. Challenge Complete the square for \begin{align*}ax^2+bx+c=0\end{align*}. Follow the steps from Examples A and B. Your final answer should be in terms of \begin{align*}a, b,\end{align*} and \begin{align*}c\end{align*}.
2. For the equation \begin{align*}8x^2+6x-5=0\end{align*}, use the formula you found in #13 to solve for \begin{align*}x\end{align*}.
3. Is the equation in #14 factorable? If so, factor and solve it.
4. Error Analysis Examine the worked out problem below.

Plug the answers into the original equation to see if they work. If not, find the error and correct it.

### Vocabulary Language: English

Perfect Square Trinomial

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression of the form $a^2+2ab+b^2$ (which can be rewritten as $(a+b)^2$) or $a^2-2ab+b^2$ (which can be rewritten as $(a-b)^2$).
Square Root

Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.