Question 1

Simplify the following expression and write the answer in standard form: $$\frac{i$$^{105}$$ - i$$^{103}$$}{i$$^{85}$$}$$

Accepted Answer

$$2$$

Question 2

A medical imaging device uses complex numbers to process signals. If the signal is represented by $$\frac{3 + i}{3 + 5i} + \frac{3 - i}{3 - 5i}$$, what is the simplified form in standard notation?

Accepted Answer

$$\frac{14}{17}$$

Question 3

Expand and write in standard form: $$(2 - i)^3$

Accepted Answer

$$-2 - 11i$$

Question 4

A chemist calculates the product of two imaginary roots from a reaction: $$\sqrt{-11} 	imes \sqrt{-7}. $$What is the result?

Accepted Answer

$$i\sqrt{77}$$

Question 5

Multiply and write the answer in standard form: $$(8 - i)(8 + i)(2 - 6i)$$

Accepted Answer

$$390 - 130i$$

Question 6

A physicist divides a constant by $$-i to $$rotate a vector in the complex plane. What is $$\frac{119}{-i} in $$standard form?

Accepted Answer

$$119i$$

Question 7

For the rational expression $$\frac{5}{x^2$ - 7x + 6}$$, which values of $$x$$ make the expression undefined?

Accepted Answer

$$\{1, 6\}$$

Question 8

Given the quadratic equation $$x^2 - 11x - 12 = 0$, calculate the discriminant and determine the number and type of solutions.

Accepted Answer

$$D = 169$$, two real solutions

Question 9

To complete the square for $$x^2 - 9x$, what term must be added, and what is the resulting perfect square trinomial factored?

Accepted Answer

$$\frac{81}{4}, (x - \frac{9}{2})^2$

Question 10

Solve using the square root property: $$k^2 - 361 = 0$

Accepted Answer

$$\pm 19$$

Question 11

Solve $$(-3x + 7)^2$ = -135$$ using the square root property. What are the solutions?

Accepted Answer

$$\frac{7}{3} + i\sqrt{15}, \frac{7}{3} - i\sqrt{15}$$

Question 12

A biologist models population growth with a quadratic equation. If the discriminant is negative, what does this imply about the population model's solutions?

Accepted Answer

There are two complex solutions

Question 13

Which of the following is the correct standard form for the product $$\sqrt{-25} 	imes \sqrt{-4}$$?

Accepted Answer

$$i\sqrt{100}$$

Question 14

A quadratic equation $$ax^2 + bx + c = 0$ has a discriminant D = 0$. What does this indicate about the roots?

Accepted Answer

One real root (double root)

Question 15

If $$i$$ represents the imaginary unit, what is $$i^4$$?

Accepted Answer

$$-1$$

Question 16

A quadratic equation models the height of a projectile: $$h(t) = -16t^2$ + 32t + 5. $$What is the discriminant for the equation $$-16t^2 + 32t + 5 = 0$?

Accepted Answer

$$32^2 - 4(-16)(5$$)$$

Question 17

Which value completes the square for $$x^2 + 8x$ and what is the resulting trinomial factored?

Accepted Answer

$$16, (x + 4)^2$

Question 18

A quadratic equation $$x^2 + 4x + 4 = 0$ is solved by completing the square. What are the roots?

Accepted Answer

$$x = -2$$

Question 19

If a quadratic equation has complex roots, which of the following must be true about its discriminant?

Accepted Answer

The discriminant is negative

Step-by-Step Guidance: Quadratic Functions and Complex Numbers (College Algebra)

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