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 $3x^2y^3 + 12x^3y^2 - 30x^4y^5$. 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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