Q: Simplify $$i^{23}.$$

$$i^{23}$$ = $$i^{4 \times 5 + 3}$$ = $$(i^4)^5$ \cdot i^3$ = 1^5$ \cdot i^3$ = i^3$ = -i$

Question 1

Which of the following is the correct way to express the complex number $$2 - 5i$$ divided by $$1 - 6i in $$standard form $$a + bi$$?

Accepted Answer

$$\frac{2 - 5i}{1 - 6i} = \frac{(2 - 5i)(1 + 6i)}{(1 - 6i)(1 + 6i)} = \frac{2 + 12i - 5i - 30i^2$}{1 - 36i^2$} = \frac{2 + 7i + 30}{1 + 36} = \frac{32 + 7i}{37} = \frac{32}{37} + \frac{7}{37}i$$

Question 2

Given the quadratic formula $$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$, what type of solutions does the equation have if $$b^2 - 4ac$$ \u003c 0$$?

Accepted Answer

The equation has two complex solutions (conjugates).

Question 3

Which of the following is the conjugate of the complex number $$7 - 4i$$?

Accepted Answer

$$7 + 4i$$

Question 4

Simplify $$i^{23}.$$

Accepted Answer

$$i^{23}$$ = $$i^{4 	imes 5 + 3}$$ = $$(i^4)^5$ \cdot i^3$ = 1^5$ \cdot i^3$ = i^3$ = -i$

Question 5

If $$z = 3 + 4i$$ and $$w = 1 - 2i$$, what is $$z + w$$?

Accepted Answer

$$4 + 2i$$

Question 6

Which of the following statements is true about the product of a complex number and its conjugate?

Accepted Answer

The product is always a real number.

Question 7

Given $$f(x) = x^2$ + 4x + 13$$, what are the zeros of $$f(x)$$?

Accepted Answer

$$x = -2 \pm 3i$$

Question 8

Which of the following graphs represents a quadratic function with complex zeros?

Accepted Answer

A graph that does not intersect the x-axis.

Question 9

What is the result of multiplying $$(5 + 2i)(5 - 2i)$$?

Accepted Answer

$$29$$

Question 10

A quadratic equation $$ax^2 + bx + c = 0$ has a discriminant b^2$ - 4ac = 0. $$What can you conclude about its solutions?

Accepted Answer

The equation has one real-number solution (a repeated root).

Question 11

If $$z = 6i$$, which of the following best describes $$z$$?

Accepted Answer

Pure imaginary number

Question 12

Which of the following is the standard form of $$\frac{3 + 2i}{1 - i}$$?

Accepted Answer

$$\frac{1}{2} + \frac{5}{2}i$$

Question 13

If $$i^2 = -1$, what is i^4$?

Accepted Answer

$$1$$

Question 14

Which of the following is a property of addition for complex numbers?

Accepted Answer

Addition is commutative: $$a + bi + c + di = c + di + a + bi$$

Study Guide: Complex Numbers and Quadratic Equations in Precalculus

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