Q: A chemist needs to mix a $20\%$ saline solution with a $50\%$ saline solution to obtain $10$ liters of a $30\%$ solution. Let $x$ be the amount (in liters) of the $20\%$ solution used. Which equation models this situation?

$0.2x + 0.5(10 - x) = 0.3 \times 10$

Question 1

Which of the following is the greatest common factor (GCF) of the polynomial $12x^3y^2 - 18x^2y^3 + 6x^4y$?

Accepted Answer

$6x^2y$

Question 2

Factor the following trinomial completely: $x^2 - 7x + 12$.

Accepted Answer

$(x-3)(x-4)$

Question 3

Which of the following is a correct factorization of the difference of squares $16x^2 - 81$?

Accepted Answer

$(4x-9)(4x+9)$

Question 4

Solve the quadratic equation $x^2 + 2x - 15 = 0$ using the zero-factor property.

Accepted Answer

$x = 3$ or $x = -5$

Question 5

A rectangular garden has an area of $60$ square meters and a length that is $4$ meters longer than its width. Which equation can be used to find the width $w$ of the garden?

Accepted Answer

$w(w+4) = 60$

Question 6

Simplify the rational expression: $\dfrac{6x^2 - 12x}{3x}$.

Question 7

Multiply and simplify: $\dfrac{x^2 - 9}{x^2 - x - 6} 	imes \dfrac{x - 2}{x + 3}$.

Accepted Answer

$\dfrac{x - 3}{x - 2}$

Question 8

Simplify the complex fraction: $\dfrac{\dfrac{2}{x}}{\dfrac{4}{x^2}}$.

Question 9

Rewrite the radical expression $\sqrt[3]{x^5}$ as an exponential expression.

Accepted Answer

$x^{5/3}$

Question 10

Solve for $x$: $\sqrt{x + 5} = 4$.

Accepted Answer

$x = 11$

Question 11

Which of the following is the simplified form of $\sqrt{50} + 2\sqrt{8}$?

Accepted Answer

$7\sqrt{2}$

Question 12

Solve the quadratic equation by completing the square: $x^2 + 6x + 5 = 0$.

Accepted Answer

$x = -1$ or $x = -5$

Question 13

A chemist needs to mix a $20\%$ saline solution with a $50\%$ saline solution to obtain $10$ liters of a $30\%$ solution. Let $x$ be the amount (in liters) of the $20\%$ solution used. Which equation models this situation?

Accepted Answer

$0.2x + 0.5(10 - x) = 0.3 	imes 10$

Question 14

Solve using the quadratic formula: $2x^2 - 4x - 6 = 0$.

Accepted Answer

$x = 3$ or $x = -1$

Question 15

Which of the following is the correct simplification of $\dfrac{x^2 - 16}{x^2 - 8x + 16}$?

Question 16

If $\sqrt{a} = 7$, what is the value of $a$?

Accepted Answer

$49$

Question 17

A car travels $300$ miles at a constant speed. If the trip takes $5$ hours, what is the speed of the car?

Accepted Answer

$60$ miles per hour

Question 18

Which of the following is the correct simplification of $\dfrac{3x^2y}{6xy^2}$?

Accepted Answer

$\dfrac{x}{2y}$

Question 19

Which of the following is a perfect square trinomial?

Accepted Answer

$x^2 + 10x + 25$

MAT 101 Final Exam Study Guide: Factoring, Rational Expressions, Radicals, and Quadratic Equations

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