The area of a parallelogram is given by the equation *x* is the length of the base. What is the length of this base?

### Completing the Square

When there is a number in front of

Let's determine the number *c* that completes the square of

Previously, we just added *a* into consideration. Let's pull out the GCF of 2 and 8 first.

Now, there is no number in front of

Add this number *inside* the parenthesis and distribute the 2.

So,

Now, let's solve the following problems by completing the square.

- Solve
3x2−9x+11=0 .

Step 1: Write the polynomial so that

Step 2: Pull out

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

To do this, divide the

Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add

Step 5: Factor the left side and simplify the right.

Step 6: Solve by using square roots.

Be careful with the addition of Step 2 and the changes made to Step 4. A very common mistake is to add

- Solve
4x2+7x−18=0 .

Let’s follow the steps from problem #1 above.

Step 1: Write the polynomial so that

Step 2: Pull out

Step 3: Now, complete the square. Find

Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add

Step 5: Factor the left side and simplify the right.

Step 6: Solve by using square roots.

### Examples

#### Example 1

Earlier, you were asked to find the length of the base of the parallelogram.

We can't factor

Step 1: Write the polynomial so that

Step 2: Pull out

Step 3: Now, complete the square. Find

Step 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*}3\left(x^2+ 3x{\color{red}+\frac{9}{4}}\right)=5{\color{red}+\frac{27}{4}}\end{align*}

Step 5: Factor the left side and simplify the right.

\begin{align*}3\left(x+\frac{3}{2}\right)^2=\frac{47}{4}\end{align*}

Step 6: Solve by using square roots.

\begin{align*}\left(x+\frac{3}{2}\right)^2 &=\frac{47}{12}\\
x+\frac{3}{2} &=\pm \frac{\sqrt{47}}{\sqrt{12}}\\
x &=-\frac{3}{2} \pm \frac{{\sqrt{47}}}{2\sqrt{3}}\\
x &=-\frac{3}{2} \pm \frac{{\sqrt{141}}}{6} \end{align*}

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*}

#### Example 2

Solve the following quadratic equation by completing the square: \begin{align*}5x^2+29x-6=0\end{align*}

\begin{align*}5x^2+29x-6 &=0\\
5\left(x^2+\frac{29}{5}x\right) &=6\\
5\left(x^2+\frac{29}{5}x+\frac{841}{100}\right) &=6+\frac{841}{20}\\
5\left(x+\frac{29}{10}\right)^2 &=\frac{961}{20}\\
\left(x+\frac{29}{10}\right)^2 &=\frac{961}{100}\\
x+\frac{29}{10} &=\pm \frac{31}{10}\\
x &=-\frac{29}{10} \pm \frac{31}{10}\\
x &=-6, \frac{1}{5}\end{align*}

#### Example 3

Solve the following quadratic by completing the square: \begin{align*}8x^2-32x+4=0\end{align*}

\begin{align*}8x^2-32x+4 &=0\\
8(x^2-4x) &=-4\\
8(x^2-4x+4) &=-4+32\\
8(x-2)^2 &=28\\
(x-2)^2 &=\frac{7}{2}\\
x-2 &=\pm \frac{\sqrt{7}}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}\\
x &=2\pm \frac{\sqrt{14}}{2}\end{align*}

### Review

Solve the quadratic equations by completing the square.

- \begin{align*}6x^2-12x-7=0\end{align*}
6x2−12x−7=0 - \begin{align*}-4x^2+24x-100=0\end{align*}
−4x2+24x−100=0 - \begin{align*}5x^2-30x+55=0\end{align*}
5x2−30x+55=0 - \begin{align*}2x^2-x-6=0\end{align*}
2x2−x−6=0 - \begin{align*}\frac{1}{2}x^2+7x+8=0\end{align*}
12x2+7x+8=0 - \begin{align*}-3x^2+4x+15=0\end{align*}
−3x2+4x+15=0

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

- \begin{align*}4x^2-4x-8=0\end{align*}
4x2−4x−8=0 - \begin{align*}2x^2+9x+7=0\end{align*}
2x2+9x+7=0 - \begin{align*}-5(x+4)^2-19=26\end{align*}
−5(x+4)2−19=26 - \begin{align*}3x^2+30x-5=0\end{align*}
3x2+30x−5=0 - \begin{align*}9x^2-15x-6=0\end{align*}
9x2−15x−6=0 - \begin{align*}10x^2+40x+88=0\end{align*}
10x2+40x+88=0

Problems 13-15 build off of each other.

**Challenge**Complete the square for \begin{align*}ax^2+bx+c=0\end{align*}ax2+bx+c=0 . Follow the steps outlined in this lesson. Your final answer should be in terms of \begin{align*}a, b,\end{align*}a,b, and \begin{align*}c\end{align*}c .- For the equation \begin{align*}8x^2+6x-5=0\end{align*}
8x2+6x−5=0 , use the formula you found in #13 to solve for \begin{align*}x\end{align*}x . - Is the equation in #14 factorable? If so, factor and solve it.
**Error Analysis**Examine the worked out problem below.

\begin{align*}4x^2-48x+11&=0\\
4(x^2-12x+\underline{\;\;\;\;\;\;}) &=-11\\
4(x^2-12x+36) &=-11+36\\
4(x-6)^2 &=25\\
(x-6)^2 &=\frac{25}{4}\\
x-6 &=\pm \frac{5}{2}\\
x &=6\pm \frac{5}{2} \rightarrow \frac{17}{2},\frac{7}{2}\end{align*}

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

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.12.