Question 1

A pharmaceutical company is modeling the concentration of a drug in the bloodstream over time using the equation $$C(t) = -2t^2$ + 8t + 10. $$After how many hours will the concentration reach its maximum, and what is that maximum concentration?

Accepted Answer

At $$t = 2$$ hours, maximum concentration is $$C(2) = 18.$$

Question 2

Which of the following equations is quadratic in form and can be solved by substitution?

Accepted Answer

$$x^4$$ - $$5x^2 + 6 = 0$

Question 3

A biologist is studying the growth of a population modeled by $$P(t) = t^2$ - 6t + 9. $$Using the discriminant, how many real solutions does $$P(t) = 0$$ have, and what does this mean biologically?

Accepted Answer

One real solution; the population reaches zero at a single time.

Question 4

Solve the radical equation $$\sqrt{x + 5} = x - 1$$ and identify any extraneous solutions.

Accepted Answer

Only $$x = 4 is $$valid; $$x = -1 is $$extraneous.

Question 5

A chemical reaction is modeled by $$2x^2 - 7x + 3 = 0$. Which method is most efficient for solving this equation, and what are the solutions?

Accepted Answer

Factoring; $$x = \frac{1}{2},\ 3$$

Question 6

A medical researcher uses the equation $$(x - 4)^2$ = 11 to $$model the spread of a virus. What are the possible values of $$x$$?

Accepted Answer

$$x = 4 \pm \sqrt{11}$$

Question 7

Which of the following is a common error when solving quadratic equations?

Accepted Answer

Applying the Zero Product Principle before setting the equation equal to zero.

Question 8

Solve the equation $$x^{2/3}$$ - $$5x^{1/3} + 6 = 0$ for real values of x$.

Accepted Answer

$$x = 8,\ 27$$

Question 9

A student solves $$x^2 = 49$ and writes x = 7$. What error did they make?

Accepted Answer

They forgot the $$\pm$$ sign; the correct answer is $$x = \pm 7.$$

Question 10

A population model uses $$x^4$$ - $$13x^2 + 36 = 0$. How many real solutions does this equation have?

Accepted Answer

Four real solutions: $$x = -3,\ -2,\ 2,\ 3$$

Question 11

Given the quadratic equation $$x^2 + 4x + 13 = 0$, what type of solutions does it have?

Accepted Answer

Two nonreal complex conjugate solutions.

Question 12

A student is asked to solve $$3x^3 - 27x = 0$. What is the correct set of solutions?

Accepted Answer

$$x = -3,\ 0,\ 3$$

Question 13

A radical equation $$\sqrt{2x + 3} + 1 = x is $$solved. Which value is valid?

Accepted Answer

$$x = 2 + \sqrt{6}$$

Question 14

A quadratic equation $$ax^2 + bx + c = 0$ has D$$ = $$b^2 - 4ac = 0$. What does this imply about its solutions?

Accepted Answer

One real solution (a repeated root).

Question 15

A student is solving $$x^2 + 8x - 9 = 0$ by completing the square. What is the correct next step after moving the constant to the other side?

Accepted Answer

Question 16

A student solves $$x^2 + 8x = 9$ by completing the square. What is the correct value to add to both sides?

Accepted Answer

Add $$16 to $$both sides.

Question 17

A quadratic equation $$2x^2 + 8x - 5 = 0$ is to be solved by completing the square. What is the first step?

Accepted Answer

Divide both sides by $$2 to $$normalize the coefficient of $$x^2.$$

Question 18

A student is solving $$x^2 + x = 20$ by factoring. What is the correct way to set up the equation?

Accepted Answer

Move all terms to one side: $$x^2 + x - 20 = 0$.

Question 19

A radical equation $$\sqrt{x + 4} + \sqrt{x} = 4 is $$solved. What is the correct value of $$x$$?

Accepted Answer

$$x = \frac{9}{4}$$

Solving Quadratic, Quadratic-in-Form, and Radical Equations: College Algebra Study Guide

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