# Lesson 16

Solving Quadratics

### Problem 1

What number should be added to the expression \(x^2 - 15x\) to result in an expression equivalent to a perfect square?

-7.5

7.5

-56.25

56.25

### Solution

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### Problem 2

Noah uses the quadratic formula to solve the equation \(2x^2+3x-5=4\). He finds \(x = \text-2.5\) or 1. But, when he checks his answer, he finds that neither -2.5 nor 1 are solutions to the equation. Here are his steps:

\(a=2\), \(b=3\), \(c=\text-5\)

\(x=\frac{\text-3 \pm \sqrt{3^2 - 4 \boldcdot 2 \boldcdot \text-5}}{2 \boldcdot 2}\)

\(x=\frac{\text-3 \pm \sqrt{49}}{4}\)

\(x = \text-2.5\) or 1

- Explain what Noah’s mistake was.
- Solve the equation correctly.

### Solution

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### Problem 3

Solve each quadratic equation with the method of your choice.

- \(x^2-2x=\text-1\)
- \(x^2+8x+14=23\)
- \(x^2-15=0\)
- \(7x^2-2x-5=0\)
- \(2x^2+12x=8\)

### Solution

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### Problem 4

What are the solutions to the equation \(x^2-4x=\text-3\)?

\(\frac{4 \pm \sqrt{16 - 4 \boldcdot 0 \boldcdot \text-3}}{2 \boldcdot 0}\)

\(\frac{4 \pm \sqrt{16 - 4 \boldcdot 1 \boldcdot \text-3}}{2 \boldcdot 1}\)

\(\frac{4 \pm \sqrt{16 - 4 \boldcdot 1 \boldcdot 3}}{2 \boldcdot 1}\)

\(\frac{\text-4 \pm \sqrt{16 - 4 \boldcdot 1 \boldcdot 3}}{2 \boldcdot 1}\)

### Solution

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### Problem 5

Which expression is equivalent to \(\sqrt{\text-23}\)?

\(\text-23i\)

\(23i\)

\(\text- i \sqrt{23}\)

\(i \sqrt{23}\)

### Solution

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(From Unit 3, Lesson 11.)### Problem 6

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

- \(5i^2\)
- \(i^2 \boldcdot i^2\)
- \((\text-3i)^2\)
- \(7 \boldcdot 4i\)
- \((5+4i) - (\text-3 + 2i)\)

### Solution

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(From Unit 3, Lesson 12.)### Problem 7

Let \(m=(7-2i)\) and \(k=3i\). Write each expression in the form \(a+bi\), where \(a\) and \(b\) are real numbers.

- \(k-m\)
- \(k^2\)
- \(m^2\)
- \(k \boldcdot m\)

### Solution

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(From Unit 3, Lesson 13.)