Free Compass Math Test Practice Questions
Solutions with Detailed Explanations – Sample 5

Solutions with full explanations to Compass math test practice questions (Sample 5).

1. The solution of the equation \(-3x + 7 = 4x - 12\) falls between what two numbers?
   
   A) \(3\) and \(3.5\)
   B) \(1\) and \(2.5\)
   C) \(2.1\) and \(3\)
   D) \(4\) and \(4.5\)
   E) \(19\) and \(7\)
   

   Solution

   \[ \begin{aligned} -3x + 7 &= 4x - 12 \\ -3x - 4x &= -12 - 7 \\ -7x &= -19 \\ x &= \frac{19}{7} \approx 2.714 \end{aligned} \] The solution \(x = \frac{19}{7}\) lies between \(2.1\) and \(3\).
   Answer: C

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2. \(35x^{2} - 11x - 6\) is the product of \(5x - 3\) and __________.

   Solution

   Factor the trinomial: \[ 35x^{2} - 11x - 6 = (5x - 3)(7x + 2) \] The second factor is \(\boxed{7x + 2}\).

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3. For all real numbers \(x, y, z\): \(\sqrt[3]{64 x^{3} y z^{2}} =\)

   Solution

   \[ \begin{aligned} \sqrt[3]{64 x^{3} y z^{2}} &= 64^{1/3} \cdot (x^{3})^{1/3} \cdot y^{1/3} \cdot (z^{2})^{1/3} \\ &= 4 \cdot x \cdot y^{1/3} \cdot z^{2/3} \\ &= 4x\,y^{1/3}z^{2/3} \end{aligned} \]

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4. For all \(x\), \(45x^{4} - 115x^{3} - 60x^{2} =\)
   
   A) \(45x^{2}(x + 4)(x - 3)\)
   B) \(5x^{2}(9x + 4)(x - 3)\)
   C) \(5x^{2}(9x - 4)(x + 3)\)
   D) \(5x^{2}(9x - 4)(x - 3)\)
   E) \(45x^{2}(x - 4)(x + 3)\)

   Solution

   \[ \begin{aligned} 45x^{4} - 115x^{3} - 60x^{2} &= 5x^{2}(9x^{2} - 23x - 12) \\ &= 5x^{2}(9x + 4)(x - 3) \end{aligned} \] Answer: B

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5. Which of the following is one of the factors of the polynomial \(-3x^{2} - 7x + 26\)?
   
   A) \(x + 2\)
   B) \(x + 1\)
   C) \(-3x + 7\)
   D) \(x - 2\)
   E) \(3x - 2\)

   Solution

   \[ -3x^{2} - 7x + 26 = (-3x - 13)(x - 2) \] One factor is \(\boxed{x - 2}\)   (option D).

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6. What is the product of the solutions of the equation \(2x^{2} + x - 15 = 0\)?

   Solution

   For a quadratic \(ax^{2} + bx + c = 0\), the product of roots equals \(\dfrac{c}{a}\). \[ \text{Product} = \frac{-15}{2} = -\frac{15}{2} \]

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7. If \(y = |x - 3|\), what is the value of \(y^{2}\) when \(x = -2\)?

   Solution

   \[ y = |-2 - 3| = |-5| = 5,\quad y^{2} = 5^{2} = 25 \]

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8. The operation \(|||\) is defined by \(x \,|||\, y = x + y + xy\). If \(-5 \,|||\, y = -13\), then \(y =\)

   Solution

   \[ \begin{aligned} -5 \,|||\, y &= (-5) + y + (-5)y = -5 + y - 5y = -5 - 4y \\ -5 - 4y &= -13 \quad\Rightarrow\quad -4y = -8 \quad\Rightarrow\quad y = 2 \end{aligned} \]

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9. A car uses 15 gallons of gas to travel 300 miles. How many gallons are needed for this car to travel 430 miles?

   Solution

   Fuel consumption rate: \(\dfrac{15 \text{ gal}}{300 \text{ mi}} = \dfrac{1}{20}\) gal/mi. \[ \text{Gallons for }430\text{ mi} = \frac{1}{20} \times 430 = 21.5 \text{ gallons} \]

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10. David has 200 books in his library. 30% of these books are about science. Of these science books, 20% are about mathematics. How many math books does David have?

    Solution

    \[ \text{Science books} = 30\% \times 200 = 60 \] \[ \text{Math books} = 20\% \times 60 = 12 \]

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11. \(\sqrt{32}\,\sqrt{2} =\)

    Solution

    \[ \sqrt{32}\,\sqrt{2} = \sqrt{32 \times 2} = \sqrt{64} = 8 \] (Alternative: \(\sqrt{32}=4\sqrt{2}\), then \(4\sqrt{2}\cdot\sqrt{2}=4\cdot 2=8\))

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12. If \(a = 3\) and \(b = -1\), evaluate \(-3a^{2} - 2ab + b^{3}\).

    Solution

    \[ \begin{aligned} -3(3)^{2} - 2(3)(-1) + (-1)^{3} &= -3(9) + 6 - 1 \\ &= -27 + 6 - 1 = -22 \end{aligned} \]

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13. \((-3x^{2} - x - 10) - (-3x + 8) =\)

    Solution

    \[ \begin{aligned} &(-3x^{2} - x - 10) - (-3x + 8) \\ &= -3x^{2} - x - 10 + 3x - 8 \\ &= -3x^{2} + 2x - 18 \end{aligned} \]

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14. \((4 - \sqrt{5})(4 + \sqrt{5}) =\)

    Solution

    \[ (4 - \sqrt{5})(4 + \sqrt{5}) = 4^{2} - (\sqrt{5})^{2} = 16 - 5 = 11 \]

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15. In a standard \((x,y)\) coordinate plane, the point \((-2 , -6)\) is located in which quadrant?

    Solution

    Both coordinates are negative, therefore the point lies in quadrant III.

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16. \(|-7 - 9| =\)

    Solution

    \[ |-7 - 9| = |-16| = 16 \]

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17. If the ratio of \(7\) to \(x\) is equal to \(\dfrac{42}{30}\), then \(x =\)

    Solution

    \[ \frac{7}{x} = \frac{42}{30} \implies 7 \times 30 = 42x \implies 210 = 42x \implies x = 5 \]

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18. If \(x = \dfrac{1}{3}\), then \(4x^{2} - 2x + 2 =\)

    Solution

    \[ \begin{aligned} 4\left(\frac{1}{3}\right)^{2} - 2\left(\frac{1}{3}\right) + 2 &= 4 \cdot \frac{1}{9} - \frac{2}{3} + 2 \\ &= \frac{4}{9} - \frac{6}{9} + \frac{18}{9} \\ &= \frac{16}{9} \end{aligned} \]

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19. Which equation corresponds to a line that is perpendicular to the line \(2y + 3x = 6\)?
    
    A) \(y = -\frac{2}{3}x + 9\)
    B) \(y = \frac{3}{2}x + 4\)
    C) \(y = -\frac{3}{2}x - 5\)
    D) \(y = \frac{2}{3}\)
    E) \(y = \frac{2}{3}x + 3\)

    Solution

    Write given line in slope-intercept form: \[ 2y = -3x + 6 \quad\Rightarrow\quad y = -\frac{3}{2}x + 3 \] Slope \(m = -\frac{3}{2}\). Perpendicular slope \(m_{\perp} = \frac{2}{3}\). The line \(y = \frac{2}{3}x + 3\) (option E) has slope \(\frac{2}{3}\).
    Answer: E

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20. \(\dfrac{\sqrt{128}}{2} + \dfrac{5\sqrt{2}}{4} =\)

    Solution

    \[ \begin{aligned} \frac{\sqrt{128}}{2} + \frac{5\sqrt{2}}{4} &= \frac{\sqrt{64 \cdot 2}}{2} + \frac{5\sqrt{2}}{4} \\ &= \frac{8\sqrt{2}}{2} + \frac{5\sqrt{2}}{4} \\ &= 4\sqrt{2} + \frac{5\sqrt{2}}{4} \\ &= \frac{16\sqrt{2}}{4} + \frac{5\sqrt{2}}{4} = \frac{21\sqrt{2}}{4} \end{aligned} \]

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