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.

A:

\(5 \pm 2\)

B:

\(\sqrt4 \pm 1\)

C:

\(\frac12 \pm 3\)

D:

\(10 \pm \sqrt3\)

E:

\(\pm \sqrt{25} \)

F:

\(1 \pm \sqrt2 \)

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.

A:

\(x^2 - 9x = \frac12\)

B:

\(x^2 + 6.4x - 8.9 = 0\)

C:

\(x^2 - 5x = 11\)

D:

\(x^2 + 0.1x + 0.0005 = 0\)

E:

\(x^2 - \frac67 x = \frac{1}{49}\)

F:

\(x^2 + 1.21x = 6.28\)

1:

\((x - 2.5)^2 = 17.25\)

2:

\((x - \frac92)^2 = \frac{83}{4}\)

3:

\((x - \frac37 )^2 = \frac{10}{49}\)

4:

\((x + 0.05)^2 = 0.002\)

5:

\((x + 3.2)^2 = 19.14\)

6:

\((x + 0.605)^2 = 6.646025\)

(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?

A:

\(x^2-6x+8\)

B:

\((x-6)^2+3\)

C:

\((x-3)(x-6)\)

D:

\((8-x)x\)

(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.)