Lesson 15

Quadratic Equations with Irrational Solutions

  • Let’s find exact solutions to quadratic equations even if the solutions are irrational.

15.1: Roots of Squares

Here are some squares whose vertices are on a grid.

3 squares labeled A, B, and C on a square grid.

Find the area and the side length each square.

square area (square units) side length (units)

15.2: Solutions Written as Square Roots

Solve each equation. Use the \(\pm\) notation when appropriate.

  1. \(x^2 - 13 = \text-12\)
  2. \((x-6)^2 = 0\)
  3. \(x^2 + 9 = 0\)
  4. \(x^2 = 18\)
  5. \(x^2 + 1 =18\)
  6. \((x + 1)^2 = 18\)

15.3: Finding Irrational Solutions by Completing the Square

Here is an example of an equation being solved by graphing and by completing the square.

Function on a grid.

\(\displaystyle \begin {align} x^2 + 6x +7 &=0\\ x^2 + 6x + 9 &= 2\\(x+3)^2 &= 2\\x+3 &=\pm \sqrt2\\ x &=\text-3\pm \sqrt2 \end{align}\)

Verify: \(\sqrt2\) is approximately 1.414. So \(\text-3+\sqrt2 \approx \text-1.586\) and \(\text-3-\sqrt2 \approx \text-4.414\).

For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close. If you get stuck, study the example.

  1. \(x^2+4x+1=0\)
  2. \(x^2-10x+18=0\)
  3. \(x^2+5x+\frac14=0\)
  4. \(x^2+\frac83 x + \frac{14}{9}=0\)

Write a quadratic equation of the form \(ax^2 + bx + c = 0\) whose solutions are  \(x = 5-\sqrt{2}\) and \(x = 5+\sqrt{2}\).


When solving quadratic equations, it is important to remember that:

  • Any positive number has two square roots, one positive and one negative, because there are two numbers that can be squared to make that number. (For example, \(6^2\) and \((\text-6)^2\) both equal 36, so 6 and -6 are both square roots of 36.)
  • The square root symbol (\(\sqrt{\phantom{3}}\)) can be used to express the positive square root of a number. For example, the square root of 36 is 6, but it can also be written as \(\sqrt{36}\) because \(\sqrt{36} \boldcdot \sqrt{36} = 36\).
  • To express the negative square root of a number, say 36, we can write -6 or \(\text- \sqrt {36}\).
  • When a number is not a perfect square—for example, 40—we can express its square roots by writing \(\sqrt{40}\) and \(\text- \sqrt{40}\).

How could we write the solutions to an equation like \((x + 4)^2 = 11\)? This equation is saying, “something squared is 11.” To make the equation true, that something must be \(\sqrt{11}\) or \(\text-\sqrt{11}\). We can write:

\(\displaystyle \begin {align} x+4 = \sqrt{11} \quad &\text{or} \quad x+4 = \text- \sqrt{11}\\x = \text-4 + \sqrt{11} \quad &\text{or} \quad x=\text-4 - \sqrt{11} \end {align}\)

A more compact way to write the two solutions to the equation is: \(x=\text-4 \pm \sqrt{11}\).

About how large or small are those numbers? Are they positive or negative? We can use a calculator to compute the approximate values of both expressions:

\(\displaystyle \text-4 + \sqrt{11} \approx \text-0.683 \quad \text{or} \quad \text-4 - \sqrt{11} \approx \text-7.317\)

We can also approximate the solutions by graphing. The equation \((x+4)^2=11\) is equivalent to \((x+4)^2-11=0\), so we can graph the function \(y=(x+4)^2-11\) and find its zeros by locating the \(x\)-intercepts of the graph.

Function on a grid.

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 \(a(x+p)^2=q\).

  • irrational number

    An irrational number is a number that is not rational. That is, it cannot be expressed as a positive or negative fraction, or zero.

  • 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.