Lesson 16

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

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

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

Solve each quadratic equation with the method of your choice.

1. \(x^2-2x=\text-1\)
2. \(x^2+8x+14=23\)
3. \(x^2-15=0\)
4. \(7x^2-2x-5=0\)
5. \(2x^2+12x=8\)

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

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

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

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

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

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

\(\text-23i\)

\(23i\)

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

\(i \sqrt{23}\)

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

1. \(5i^2\)
2. \(i^2 \boldcdot i^2\)
3. \((\text-3i)^2\)
4. \(7 \boldcdot 4i\)
5. \((5+4i) - (\text-3 + 2i)\)

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

1. \(k-m\)
2. \(k^2\)
3. \(m^2\)
4. \(k \boldcdot m\)