Lesson 18

The Quadratic Formula and Complex Solutions

Problem 1

Clare solves the quadratic equation \(4x^2+12x+58=0\), but when she checks her answer, she realizes she made a mistake. Explain what Clare's mistake was.

\(\begin{align} x &= \frac{\text-12 \pm \sqrt{12^2 - 4 \boldcdot 4 \boldcdot 58}}{2 \boldcdot 4} \\ x &= \frac{\text-12 \pm \sqrt{144-928}}{8} \\ x &= \frac{\text-12 \pm \sqrt{\text-784}}{8} \\ x &= \frac{\text-12 \pm 28i}{8} \\ x &= \text-1.5 \pm 28i \\ \end{align}\)

Solution

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Problem 2

Write in the form \(a+bi\), where \(a\) and \(b\) are real numbers:

  1. \(\frac{5 \pm \sqrt{\text-4}}{3}\)
  2. \(\frac{10 \pm \sqrt{\text-16}}{2}\)
  3. \(\frac{\text-3 \pm \sqrt{\text-144}}{6}\)

Solution

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Problem 3

Priya is using the quadratic formula to solve two different quadratic equations.

For the first equation, she writes \(x= \frac{4 \pm \sqrt{16-72}}{12}\)

For the second equation, she writes \(x=\frac{8 \pm \sqrt{64-24}}{6}\)

Which equation(s) will have real solutions? Which equation(s) will have non-real solutions? Explain how you know.

Solution

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Problem 4

Find the exact solution(s) to each of these equations, or explain why there is no solution.

  1. \(x^2=25\)
  2. \(x^3 = 27\)
  3. \(x^2=12\)
  4. \(x^3=12\)

Solution

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(From Unit 3, Lesson 8.)

Problem 5

Kiran is solving the equation \(\sqrt{x+2} - 5 = 11\) and decides to start by squaring both sides. Which equation results if Kiran squares both sides as his first step?

A:

\(x + 2 - 25 = 121\)

B:

\(x + 2 + 25 = 121\)

C:

\(x+2 - 10\sqrt{x+2} + 25 = 121\)

D:

\(x+2 + 10\sqrt{x+2} + 25 = 121\)

Solution

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(From Unit 3, Lesson 9.)

Problem 6

Plot each number on the real or imaginary number line.

  1. \(\text-\sqrt{4}\)
  2. \(\sqrt{\text-1}\)
  3. \(3\sqrt{4}\)
  4. \(\text-3\sqrt{\text-1}\)
  5. \(4\sqrt{\text-1}\)
  6. \(2\sqrt{2}\)
Coordinate plane. Horizontal axis, -6 to 6, by 2’s. Vertical axis, -6 times the square root of -1 to 6 times the square root of -1, by 2 times the square root of -1’s.

Solution

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(From Unit 3, Lesson 10.)