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Completing the Square when the Leading Coefficient Equals 1

Complete the perfect square to solve simple trinomials

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Completing the Square When the Leading Coefficient Equals 1

The area of a parallelogram is given by the equation \begin{align*}x^2 + 8x - 5 = 0\end{align*}, where x is the length of the base. What is the length of this base?

Completing the Square

Completing the square is another technique used to solve quadratic equations. When completing the square, the goal is to make a perfect square trinomial and factor it.

Solve the following problems

Solve \begin{align*}x^2-8x-1=10\end{align*}.

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. This is only for organizational purposes, but it really helps. Leave a little space after the \begin{align*}x-\end{align*}term.

\begin{align*}x^2-8x=11\end{align*}

2. Now, "complete the square." Determine what number would make a perfect square trinomial with \begin{align*}x^2-8x+c\end{align*}. To do this, divide the \begin{align*}x-\end{align*}term by 2 and square that number, or \begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.

\begin{align*}\left(\frac{b}{2}\right)^2=\left(\frac{8}{2}\right)^2=4^2=16\end{align*}

3. Add this number to both sides in order to keep the equation balanced.

\begin{align*}x^2-8x {\color{red}+16}=11 {\color{red}+16}\end{align*}

4. Factor the left side to the square of a binomial and simplify the right.

\begin{align*}(x-4)^2=27\end{align*}

5. Solve by using square roots.

\begin{align*}x-4 &=\pm3\sqrt{3}\\ x &=4 \pm 3 \sqrt{3}\end{align*}

Completing the square enables you to solve any quadratic equation using square roots. Through this process, we can make an unfactorable quadratic equation solvable, like the one above. It can also be used with quadratic equations that have imaginary solutions.

Solve \begin{align*}x^2+12x+37=0\end{align*}

First, this is not a factorable quadratic equation. Therefore, the only way we know to solve this equation is to complete the square. Follow the steps from Example A.

1. Organize the polynomial, \begin{align*}x\end{align*}'s on the left, constant on the right.

\begin{align*}x^2+12x=-37\end{align*}

2. Find \begin{align*}\left(\frac{b}{2}\right)^2\end{align*} and add it to both sides.

\begin{align*}\left(\frac{b}{2}\right)^2 = \left(\frac{12}{2}\right)^2 &=6^2={\color{red}36}\\ x^2+12x+{\color{red}36} &=-37+{\color{red}36}\end{align*}

3. Factor the left side and solve. \begin{align*}(x+6)^2 &=-1\\ x+6 &=\pm i\\ x &=-6\pm i\end{align*}

Solve \begin{align*}x^2-11x-15=0\end{align*}.

This is not a factorable equation. Use completing the square.

1. Organize the polynomial, \begin{align*}x\end{align*}’s on the left, constant on the right.

\begin{align*}x^2-11x=15\end{align*}

2. Find \begin{align*}\left(\frac{b}{2}\right)^2\end{align*} and add it to both sides.

\begin{align*}\left(\frac{b}{2}\right)^2 &=\left(\frac{11}{2}\right)^2 = {\color{red}\frac{121}{4}}\\ x^2-11x+{\color{red}\frac{121}{4}} &=15+{\color{red}\frac{121}{4}}\end{align*}

3. Factor the left side and solve.

\begin{align*}\left(x-\frac{11}{2}\right)^2 &=\frac{60}{4}+{\color{red}\frac{121}{4}}\\ \left(x-\frac{11}{2}\right)^2 &=\frac{181}{4}\\ x-\frac{11}{2} &=\pm \frac{\sqrt{181}}{2}\\ x &=\frac{11}{2}\pm \frac{\sqrt{181}}{2}\end{align*}

Examples

Example 1

Earlier, you were asked what is the length of the base. 

We can't factor \begin{align*}x^2 + 8x - 5 = 0\end{align*}, so we must complete the square.

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 are on the right.

\begin{align*}x^2 + 8x = 5\end{align*}

2. Now, complete the square. \begin{align*}\left(\frac{b}{2}\right)^2=\left(\frac{8}{2}\right)^2=4^2=16\end{align*}

3. Add this number to both sides in order to keep the equation balanced.

\begin{align*}x^2 + 8x {\color{red} + 16}=5 {\color{red} + 16}\end{align*}

4. Factor the left side to the square of a binomial and simplify the right.

\begin{align*}(x + 4)^2=21\end{align*}

5. Solve by using square roots.

\begin{align*}x + 4 &=\pm\sqrt{21}\\ x &= -4 \pm \sqrt{21}\end{align*}

However, because x is the length of the parallelogram's base, it must be a positive value. Only \begin{align*}-4 + \sqrt{21}\end{align*} results in a positive value. Therefore, the length of the base is \begin{align*}-4 + \sqrt{21}\end{align*}.

Example 2

Find the value of \begin{align*}c\end{align*} that would make \begin{align*}x^2-2x+c\end{align*} a perfect square trinomial. Then, factor the trinomial.

\begin{align*}c=\left(\frac{b}{2}\right)^2=\left(\frac{2}{2}\right)^2=1^2=1\end{align*}. The factors of \begin{align*}x^2-2x+1\end{align*} are \begin{align*}(x - 1)(x - 1)\end{align*} or \begin{align*}(x - 1)^2\end{align*}.

Example 3

Solve the following quadratic equations by completing the square.

\begin{align*}x^2+10x+21=0\end{align*}

Use the steps from the examples above.

\begin{align*}x^2+10x+21 &=0\\ x^2+10x &=-21\\ x^2+10x+\left(\frac{10}{2}\right)^2 &=-21+\left(\frac{10}{2}\right)^2\\ x^2+10x+25 &=-21+25\\ (x+5)^2 &=4\\ x+5 &=\pm 2\\ x &=-5\pm 2\\ x &=-7,-3\end{align*}

Example 4

\begin{align*}x-5x=12\end{align*}

Use the steps from the examples above.

\begin{align*}x^2-5x &=12\\ x^2-5x+\left(\frac{5}{2}\right)^2 &=12+\left(\frac{5}{2}\right)^2\\ x^2-5x+\frac{25}{4} &=\frac{48}{4}+\frac{25}{4}\\ \left(x-\frac{5}{2}\right)^2 &=\frac{73}{4}\\ x-\frac{5}{2} &=\pm \frac{\sqrt{73}}{2}\\ x &=\frac{5}{2}\pm \frac{\sqrt{73}}{2}\end{align*}

Review

Determine the value of \begin{align*}c\end{align*} that would complete the perfect square trinomial.

  1. \begin{align*}x^2+4x+c\end{align*}
  2. \begin{align*}x^2-2x+c\end{align*}
  3. \begin{align*}x^2+16x+c\end{align*}

Rewrite the perfect square trinomial as a square of a binomial.

  1. \begin{align*}x^2+6x+9\end{align*}
  2. \begin{align*}x^2-7x+\frac{49}{4}\end{align*}
  3. \begin{align*}x^2-\frac{1}{2}x+\frac{1}{16}\end{align*}

Solve the following quadratic equations by completing the square.

  1. \begin{align*}x^2+6x-15=0\end{align*}
  2. \begin{align*}x^2+10x+29=0\end{align*}
  3. \begin{align*}x^2-14x+9=-60\end{align*}
  4. \begin{align*}x^2+3x+18=-2\end{align*}
  5. \begin{align*}x^2-9x-5=23\end{align*}
  6. \begin{align*}x^2-20x=60\end{align*}

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

  1. \begin{align*}x^2+x-30=0\end{align*}
  2. \begin{align*}x^2-18x+90=0\end{align*}
  3. \begin{align*}x^2+15x+56=0\end{align*}
  4. \begin{align*}x^2+3x-24=12\end{align*}
  5. \begin{align*}(x-2)^2-20=-45\end{align*}
  6. \begin{align*}x^2+24x+44=-19\end{align*}
  7. Solve \begin{align*}x^2+7x-44=0\end{align*} by factoring and completing the square. Which method do you prefer?
  8. Challenge Solve \begin{align*}x^2+\frac{17}{8}x-2=-9\end{align*}.

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.11. 

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Vocabulary

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.

Completing the Square

Completing the square is a common method for rewriting quadratics. It refers to making a perfect square trinomial by adding the square of 1/2 of the coefficient of the x term.

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