Solutions with detailed explanations to compass math test practice questions in sample 5.

The solution of the equation  3x + 7 = 4x  12 falls bewteen what 2 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
We first solve the given equation
3x + 7 = 4x  12
3x  4x = 12  7
7x = 19
x = 19/7 ≈ 2.71
The solution x = 19/7 is between 2.1 and 3. Answer C

35x^{2} 11x  6 is the product of 5x  3 and
Solution
We are given one factor 5x  3 and we need to find the second factor of 35x^{2} 11x  6 as follows
35x^{2} 11x  6 = (5x  3)(7x + 2)
So 35x^{2} 11x  6 is the product of 5x  3 and 7x + 2

For all real numbers x, y and z, ^{3}√(64 x^{3} y z^{2}) =
Solution
Let us rewrite the given expression as follows
^{3}√(64 x^{3} y z^{2}) = 64^{1/3} (x^{3})^{1/3} y^{1/3} (z^{2})^{1/3}
then simplify
= 4 x y^{1/3} z^{2/3}

For all x, 45 x^{4}  115 x^{3}  60 x^{2} =
A) 45 x^{2}(x + 4)(x  3)
B) 5 x^{2}(9x + 4)(x  3)
C) 5 x^{2}(9x  4)(x + 3)
D) 5 x^{2}(9x  4)(x  3)
E) 45 x^{2}(x  4)(x + 3)
Solution
Factor 5 x^{2} out
45 x^{4}  115 x^{3}  60 x^{2} = 5x^{2} (9x^{2} 23x  12)
We now factor the trinominal
= 5x^{2} (9x + 4)(x  3)

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
Factor 3x^{2}  7x + 26
3x^{2}  7x + 26 = (3x  13)(x  2)
x  2 is a factor of the given expressions

What is the product of the solutions of the equation 2x^{2} + x  15?
Solution
The product of the solutions of a quadratic equation of the form ax^{2} + bx + c = 0 is equal to c / a. Hence the product of the solutions of the solutions of the given equation is equal to
15/2

If y = x  3, what is the value of y^{2} when x = 2?
Solution
Evaluate y and then y^{2}
y = 2  3 = 5
y^{2} = 5^{2} = 25

The operation  is defined by x  y = x + y + xy. If 5  y = 13, then y =
Solution
Use definition to express 5  y in terms of y
5  y = (5) + y + (5)y = 5 + y  5y = 5  4y
Use the fact that 5  y = 13 to write an equation
5  4y =  13
Solve for y
y = 2

A car uses 15 gallons of gaz to travel 300 miles. How many gallons are needed for this car to travel a distance of 430 miles?
Solution
Let us first find how many gallons are used to travel 1 mile
15 gallons / 300 miles = 1 /20 gallons per mile
The number of gallons needed for 430 miles is given by
(1 /20 gallons per mile)*(430 miles) = 21.5 gallons

David has 200 books in his library. 30% of these books are about science. Of these books about science, 20% are about mathematics. How many math books does David have?
Solution
Number of science books is
30% of 200 = 30% * 200 = 60 books
Number of math books is
20% of 60 = 20% * 60 = 12 bokks.

√32 √2 =
Solution
Use 32 = 2 * 16 and simplify
√32 √2 = √(2*16)√2
= √2 √16 √2
= 4 √2 √2 = 4 * 2 = 8

If a = 3 and b = 1, then 3a^{2}  2ab + b^{3}
Solution
Substitute
3a^{2}  2ab + b^{3} = 3(3)^{2}  2(3)(1) + (1)^{3}
Simplify
= 3(9) + 6  1 = 27 + 6  1 = 22

(3x^{2}  x  10)  ( 3x + 8)?
Solution
Multiply, group like terms and simplify
(3x^{2}  x  10)  ( 3x + 8) = 3x^{2}  x  10 + 3x  8)
= 3x^{2} + 2x  18

(4  √5)(4 + √5) =
Solution
Expand and simplify
(4  √5)(4 + √5) = 4^{2}  √5 ^{2} = 16  5 = 11

In a standard (x,y) coordinate plane, the point (2 , 6) is located
Solution
Both coordinates are negantive, therefore the point (2 , 6) is located in quadrant III

 7  9  =
Solution
Simplify as follows
 7  9  = 16 = 16

If the ratio of 7 to x is equal to 42/30, then x =
Solution
We are given
7 / x = 42 / 30
Solve for x
7 * 30 = 42 * x
x = 5

If x = 1/3, then 4 x^{2}  2x + 2 =
Solution
Substitute
4 x^{2}  2x + 2 = 4 (1/3)^{2}  2(1/3) + 2 =
Simplify using common denominator
4 * (1/9)  2/3 + 2 = 4 / 9  6 / 9 + 18 /9 = 16 / 9

Which equations corresponds to a line that is perpendicular to the line 2y + 3x = 6
A) y = (2/3)x + 9
B) y = (3/2)x + 4
C) y = (3/2)x  5
D) y = 2/3
E) y = (2/3)x + 3
Solution
Write in slope intercept form and find the slope of given line
2y + 3x = 6
2y = 3x + 6
y = (3/2) x + 2 , slope =  3 / 2
Slope of line perpendicular to the given line is equal to
1 / (3/2) = 2 / 3
Hence line E) given by y = (2/3)x + 3
is perpendicular to the given line

(√128) / 2 + (5√2) / 4 =
Solution
Use 128 = 2 * 64 to write
(√128) / 2 + (5√2) / 4 = (√(2*64)) / 2 + (5√2) / 4 =
= (√2 √64) / 2 + (5√2) / 4
= 8 √2 / 2 + 5√2 / 4
= 16 √2 / 4 + 5√2 / 4 = 21 √2 / 4
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