Question 1

A chemist is mixing two solutions, one with a concentration of $$x$$ grams per liter and another with $$y$$ grams per liter. If the total concentration after mixing is modeled by the equation $$13x - 9 = 6(2x + 3) - 9$$, what is the value of $$x$$?

Accepted Answer

$$x = 2$$

Question 2

A medical researcher is analyzing a dosage equation: $$\frac{1}{2}(2x + 3) = \frac{1}{3}(x - 5). $$What is the value of $$x$$ that satisfies this equation?

Accepted Answer

$$x = -3$$

Question 3

A business calculates its profit using the equation $$6x + 4 - 9(x + 1) = 8x + 3. $$What is the value of $$x$$ that makes the profit zero?

Accepted Answer

$$x = -7$$

Question 4

A construction company uses the equation $$2(5 + 2x) - 3x = 15 + 3x to $$determine the number of hours needed for a project. What is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 5

A biologist models population growth with the equation $$3x^2y^3$ + 12x^3$y^2$$ - $$30x^4y^5$. Which of the following is the greatest common factor (GCF) of the terms?

Accepted Answer

$$3x^2y^2$

Question 6

A projectile's height is modeled by $$x^2 + 13x - 48 = 0$. What are the possible values of x$ when the projectile hits the ground?

Accepted Answer

$$x = -16, x = 3$$

Question 7

A company’s profit function is $$m^2 + 16m + 45 = 0$. What are the values of m$ where the profit is zero?

Accepted Answer

$$m = -5, m = -9$$

Question 8

A quadratic equation $$14x^2 + 37x + 5 = 0$ models the path of a thrown object. Which of the following is a solution for x$?

Accepted Answer

$$x = -\frac{5}{14}$$ and $$x = -\frac{1}{14}$$

Question 9

A ball is thrown and its height is given by $$t^2 - 121 = 0$. At what times does the ball reach a height of zero?

Accepted Answer

$$t = 11, t = -11$$

Question 10

A quadratic equation $$x^2 - 17x + 72 = 0$ models the break-even points for a business. What are the solutions for x$?

Accepted Answer

$$x = 8, x = 9$$

Question 11

A quadratic equation $$15y^2 + 47y = -28$ is used to model a chemical reaction. What are the solutions for y$?

Accepted Answer

$$y = -4, y = \frac{7}{15}$$

Question 12

A quadratic equation $$x^2 - 10x + 15 = 0$ models the time it takes for a medicine to reach a certain concentration. What are the solutions for x$?

Accepted Answer

$$x = 5 \pm \sqrt{15}$$

Question 13

A quadratic equation $$-9x^2 - 8x + 7 = 0$ is used to model a physics experiment. What is the value of x$ that satisfies the equation?

Accepted Answer

$$x = \frac{-8 \pm \sqrt{64 - 4(-9)(7)}}{-18}$$

Question 14

Solve for $$x in $$the equation $$x^2 = 25$, which could represent the distance from a central point in a city grid.

Accepted Answer

$$x = -5, x = 5$$

Question 15

Calculate $$i^3$$, where $$i is $$the imaginary unit, as used in electrical engineering.

Accepted Answer

$$-i$$

Question 16

Simplify $$\sqrt{-3} \cdot \sqrt{-75}$$ using properties of complex numbers.

Accepted Answer

$$15i$$

Question 17

Simplify $$-5\sqrt{27}i$$ using properties of complex numbers.

Accepted Answer

$$-15i\sqrt{3}$$

Question 18

Simplify $$(4 - 3i)^2$ using properties of complex numbers.

Accepted Answer

$$7 - 24i$$

Question 19

A circuit has two impedances, $$8 + 7i$$ and $$6 + 10i. $$What is their sum?

Accepted Answer

$$14 + 17i$$

Question 20

Simplify $$\frac{3 + 5i}{5 + 6i}$$ and write your answer in the form $$a + bi.$$

Accepted Answer

$$\frac{28 + 2i}{61}$$

College Algebra Study Guide: Linear Equations, Quadratic Equations, and Complex Numbers

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