Lesson 20

Rational and Irrational Solutions

  • Let’s consider the kinds of numbers we get when solving quadratic equations.

Problem 1

Decide whether each number is rational or irrational.

  • 10
  • \(\frac45 \)
  • \(\sqrt4 \)
  • \(\sqrt{10}\)
  • -3
  • \(\sqrt{\frac{25}{4}}\)
  • \(\sqrt{0.6}\)

Problem 2

Here are the solutions to some quadratic equations. Select all solutions that are rational.


\(5 \pm 2\)


\(\sqrt4 \pm 1\)


\(\frac12 \pm 3\)


\(10 \pm \sqrt3\)


\(\pm \sqrt{25} \)


\(1 \pm \sqrt2 \)

Problem 3

Solve each equation. Then, determine if the solutions are rational or irrational.

  1. \((x+1)^2 = 4\)
  2. \((x-5)^2 = 36\)
  3. \((x+3)^2 = 11\)
  4. \((x-4)^2 = 6\)

Problem 4

Here is a graph of the equation \(y=81(x-3)^2-4\).

  1. Based on the graph, what are the solutions to the equation \(81(x-3)^2=4\)?

    Parabola facing up. Vertex = 3 comma -4. X intercepts at 2 point 7,7,8 and 3 point 2,2,2
  2. Can you tell whether they are rational or irrational? Explain how you know.
  3. Solve the equation using a different method and say whether the solutions are rational or irrational. Explain or show your reasoning.

Problem 5

Match each equation to an equivalent equation with a perfect square on one side.

(From Unit 7, Lesson 13.)

Problem 6

To derive the quadratic formula, we can multiply \(ax^2+bx+c=0\) by an expression so that the coefficient of \(x^2\) is a perfect square and the coefficient of \(x\) is an even number.

  1. Which expression, \(a\), \(2a\), or \(4a\), would you multiply \(ax^2+bx+c=0\) by to get started deriving the quadratic formula?
  2. What does the equation \(ax^2+bx+c=0\) look like when you multiply both sides by your answer?
(From Unit 7, Lesson 19.)

Problem 7

Here is a graph that represents \(y=x^2\).

On the same coordinate plane, sketch and label the graph that represents each equation:

  1. \(y=\text-x^2-4\)
  2. \(y=2x^2+4\)
A curve in an x y plane, origin O.
(From Unit 6, Lesson 12.)

Problem 8

Which quadratic expression is in vertex form?









(From Unit 6, Lesson 15.)

Problem 9

Function \(f\) is defined by the expression \(\frac{5}{x-2}\).

  1. Evaluate \(f(12)\).
  2. Explain why \(f(2)\) is undefined.
  3. Give a possible domain for \(f\).
(From Unit 4, Lesson 10.)