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5.11: Completing the Square When the Leading Coefficient Equals 1

Difficulty Level: At Grade Created by: CK-12
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The area of a parallelogram is given by the equation \begin{align*}x^2 + 8x - 5 = 0\end{align*}x2+8x5=0, where x is the length of the base. What is the length of this base?

Watch This

Watch the first part of this video, until about 5:25.

Khan Academy: Solving Quadratic Equations by Completing the Square

Guidance

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.

Example A

Solve \begin{align*}x^2-8x-1=10\end{align*}x28x1=10.

Solution:

1. Write the polynomial so that \begin{align*}x^2\end{align*}x2 and \begin{align*}x\end{align*}x 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*}xterm.

\begin{align*}x^2-8x=11\end{align*}x28x=11

2. Now, “complete the square.” Determine what number would make a perfect square trinomial with \begin{align*}x^2-8x+c\end{align*}x28x+c. To do this, divide the \begin{align*}x-\end{align*}xterm 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.

Example B

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

Solution: 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*}

Example C

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

Solution: 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*}

Intro Problem Revisit 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*}.

Guided Practice

1. 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.

Solve the following quadratic equations by completing the square.

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

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

Answers

1. \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*}.

2. 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*}

3. 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*}

Vocabulary

Binomial
A mathematical expression with two terms.
Square of a Binomial
A binomial that is squared.
Complete the Square
The process used to solve unfactorable quadratic equations.

Practice

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*}.

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Vocabulary

TermDefinition
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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Date Created:
Mar 12, 2013
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MAT.ALG.914.2.L.1