Lesson 12

Completing the Square (Part 1)

  • Let’s learn a new method for solving quadratic equations.

12.1: Perfect or Imperfect?

Select all expressions that are perfect squares. Explain how you know.

  1. \((x+5)(5+x)\)
  2. \((x+5)(x-5)\)
  3. \((x-3)^2\)
  4. \(x-3^2\)
  5. \(x^2+8x+16\)
  6. \(x^2+10x+20\)

12.2: Building Perfect Squares

Complete the table so that each row has equivalent expressions that are perfect squares. 

standard form factored form
1. \(x^2+6x+9\)  
2. \(x^2-10x+25\)  
3.                                   \((x-7)^2\)
4. \(x^2-20x+\underline{\hspace{0.5in}}\) \((x- \underline{\hspace{0.5in}})^2\)
5. \(x^2+16x+\underline{\hspace{0.5in}}\) \((x+ \underline{\hspace{0.5in}})^2\)
6. \(x^2+7x+\underline{\hspace{0.5in}}\) \((x+ \underline{\hspace{0.5in}})^2\)
7. \(x^2+bx+\underline{\hspace{0.5in}}\) \((x+ \underline{\hspace{0.5in}})^2\)

12.3: Dipping Our Toes in Completing the Square

One technique for solving quadratic equations is called completing the square. Here are two examples of how Diego and Mai completed the square to solve the same equation. 

Diego:

\(\displaystyle \begin {align} x^2+10x+9 &=0 \\x^2+10x &= \text-9 \\ x^2+10x+25 &=\text-9 + 25\\x^2+10x+25 &=16 \\ (x+5)^2 &=16\\ x+5=4 \quad & \text{or} \quad x+5=\text-4\\ x=\text-1 \quad & \text{or} \quad x=\text-9 \end{align}\)

Mai:

\(\begin {align} x^2 + 10x + 9 &= 0\\ x^2 + 10x + 9 + 16 &= 16\\ x^2+10x+25 &=16\\ (x+5)^2&=16\\ x+5=4 \quad & \text{or} \quad x+5=\text-4\\ x=\text-1 \quad & \text{or} \quad x=\text-9 \end {align}\)

Study the worked examples. Then, try solving these equations by completing the square:

  1. \(x^2+6x+8=0\)
  2. \(x^2+12x=13\)
  3. \(0=x^2-10x+21\)
  4. \(x^2-2x+3=83\)
  5. \(x^2+40=14x\)


Here is a diagram made of a square and two congruent rectangles. Its total area is \(x^2+35x\) square units.

A square and two congruent rectangles. Square has side length x units.
  1. What is the length of the unlabeled side of each of the two rectangles?
  2. If we add lines to make the figure a square, what is the area of the entire figure?
  3. How is the process of finding the area of the entire figure like the process of building perfect squares for expressions like \(x^2 + bx\)?

Summary

Turning an expression into a perfect square can be a good way to solve a quadratic equation. Suppose we wanted to solve \(x^2 - 14x +10 = \text-30\).

The expression on the left, \(x^2 - 14x +10\), is not a perfect square, but \(x^2 - 14x + 49\) is a perfect square. Let’s transform that side of the equation into a perfect square (while keeping the equality of the two sides).

  • One helpful way to start is by first moving the constant that is not a perfect square out of the way. Let’s subtract 10 from each side:​​​​​​

\(\displaystyle \begin {align} x^2 - 14x +10 - 10 &= \text-30 - 10\\ x^2 - 14x &= \text-40 \end {align}\)

  • And then add 49 to each side:​​​​​​

\(\displaystyle \begin {align} x^2 - 14x +49 &= \text-40 +49\\ x^2 - 14x+49 &= 9 \end {align}\)

  • The left side is now a perfect square because it’s equivalent to \((x-7)(x-7)\) or \((x-7)^2\). Let’s rewrite it:

\(\displaystyle (x-7)^2=9\)

  • If a number squared is 9, the number has to be 3 or -3. To finish up:​​​​​​

\(\displaystyle \begin {align} x-7=3 \quad & \text{or} \quad x-7=\text-3\\ x=10 \quad & \text{or} \quad x=4 \end{align}\)

This method of solving quadratic equations is called completing the square. In general, perfect squares in standard form look like \(x^2 + bx + \left(\frac{b}{2} \right)^2\), so to complete the square, take half of the coefficient of the linear term and square it.

In the example, half of -14 is -7, and \((\text-7)^2\) is 49. We wanted to make the left side \(x^2 - 14x + 49.\) To keep the equation true and maintain equality of the two sides of the equation, we added 49 to each side.

Glossary Entries

  • completing the square

    Completing the square in a quadratic expression means transforming it into the form \(a(x+p)^2-q\), where \(a\), \(p\), and \(q\) are constants.

    Completing the square in a quadratic equation means transforming into the form \(a(x+p)^2=q\).

  • perfect square

    A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.

  • rational number

    A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into \(b\) equal parts and finding the point that is \(a\) of them from 0. We can always write a fraction in the form \(\frac{a}{b}\) where \(a\) and \(b\) are whole numbers, with \(b\) not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as \(\frac{7}{10}\).

    The numbers \(3\), \(\text-\frac34\), and \(6.7\) are all rational numbers. The numbers \(\pi\) and \(\text-\sqrt{2}\) are not rational numbers, because they cannot be written as fractions or their opposites.