ASVAB Mathematics Knowledge Test 3 — Questions and Answers

Question 1: Solve for x: 3x² - 12 = 0

x = ±4
x = ±3
x = ±2 (Correct answer)
x = ±6

Correct answer: x = ±2

3x² = 12 → x² = 4 → x = ±2.

To solve 3x² - 12 = 0: add 12 to both sides → 3x² = 12; divide both sides by 3 → x² = 4; take the square root of both sides → x = ±2 (both positive and negative roots satisfy the equation since (-2)² = 4 and (2)² = 4). This quadratic has two real solutions because the discriminant b² - 4ac = 0 - 4(3)(-12) = 144 > 0. On the ASVAB MK section, quadratic equations appear regularly. Methods to solve: factoring, completing the square, or the quadratic formula x = (-b ± √(b²-4ac))/(2a). For simple equations like this one with no linear term (b=0), isolating x² and taking the square root is fastest. Always remember ± when taking square roots of both sides.

Question 2: What is the area of a triangle with base 14 cm and height 9 cm?

126 cm²
63 cm² (Correct answer)
252 cm²
94.5 cm²

Correct answer: 63 cm²

A = ½ × 14 × 9 = ½ × 126 = 63 cm².

The area formula for a triangle is A = ½ × b × h, where b is the base and h is the perpendicular height (altitude). A = ½ × 14 × 9 = ½ × 126 = 63 cm². The most common mistake is forgetting the ½ factor — answer A (126 cm²) represents b × h without the ½. Answer C (252 cm²) is 2bh. This formula works for all triangles regardless of shape: the height must be perpendicular to the base, which may require an imaginary line outside an obtuse triangle. On the ASVAB MK section, memorize these key area formulas: rectangle = lw; triangle = ½bh; circle = πr²; trapezoid = ½(b₁+b₂)h; parallelogram = bh. Also know perimeter = sum of all sides.

Question 3: Simplify: (x³ · x⁴) / x²

x⁵ (Correct answer)
x⁹
x⁴
x²⁴

Correct answer: x⁵

x³ · x⁴ = x⁷. Then x⁷ ÷ x² = x^(7-2) = x⁵.

Exponent rules are fundamental to the ASVAB MK section. The key laws: Product rule: xᵃ · xᵇ = x^(a+b). Quotient rule: xᵃ / xᵇ = x^(a-b). Power rule: (xᵃ)ᵇ = x^(ab). Zero exponent: x⁰ = 1 (for x ≠ 0). Negative exponent: x^(-n) = 1/xⁿ. Applying these: x³ · x⁴ = x^(3+4) = x⁷; then x⁷ / x² = x^(7-2) = x⁵. Trap answer x⁹ results from adding all three exponents (3+4+2) without subtracting the denominator. Trap answer x²⁴ results from multiplying all exponents. Always apply rules step by step: multiply first (add exponents), then divide (subtract denominator exponent).

Question 4: A circle has a radius of 7 cm. What is its circumference? (Use π ≈ 3.14)

21.98 cm
43.96 cm (Correct answer)
153.86 cm
87.92 cm

Correct answer: 43.96 cm

C = 2πr = 2 × 3.14 × 7 = 43.96 cm.

Circle formulas: Circumference = 2πr = πd (where d = diameter = 2r). Area = πr². With r = 7: Circumference = 2 × 3.14 × 7 = 43.96 cm. Area = 3.14 × 49 = 153.86 cm² (answer C — a common trap if you use the area formula when circumference is asked). Answer A (21.98 cm) = πr (missing the factor of 2). Answer D (87.92 cm) = 4πr. On the ASVAB, circumference vs. area confusion is a frequently used distractor strategy. Mnemonic: Circumference = 'around' the circle (linear measurement, units in cm); Area = 'inside' the circle (square measurement, units in cm²). Know both formulas cold.

Question 5: Factor completely: x² - 9

(x - 3)(x - 3)
(x + 3)(x + 3)
(x - 3)(x + 3) (Correct answer)
Cannot be factored

Correct answer: (x - 3)(x + 3)

x² - 9 = x² - 3² = (x - 3)(x + 3).

The difference of squares pattern is one of the most important factoring patterns: a² - b² = (a - b)(a + b). Verify by expanding: (x-3)(x+3) = x² + 3x - 3x - 9 = x² - 9 ✓. Common factoring patterns to know for ASVAB: Difference of squares: a² - b² = (a-b)(a+b). Perfect square trinomial: a² + 2ab + b² = (a+b)², a² - 2ab + b² = (a-b)². Sum of cubes: a³ + b³ = (a+b)(a²-ab+b²). General trinomial: x² + bx + c = (x+p)(x+q) where p+q=b and p×q=c. Answer A is (x-3)² = x²-6x+9, not x²-9. Always check by expanding your factored answer.

Question 6: What is the slope of a line passing through points (2, 5) and (6, 13)?

½
3
2 (Correct answer)
4

Correct answer: 2

m = (13 - 5) / (6 - 2) = 8 / 4 = 2.

Slope (m) measures the steepness and direction of a line and is calculated as: m = (y₂ - y₁) / (x₂ - x₁) = rise / run. Using (2,5) and (6,13): m = (13-5)/(6-2) = 8/4 = 2. The slope of 2 means for every 1 unit moved right on the x-axis, y increases by 2 units. A positive slope means the line goes up left-to-right; negative slope goes down; zero slope is horizontal; undefined slope is vertical. Once you have the slope, the line's equation is: y - y₁ = m(x - x₁) → y - 5 = 2(x - 2) → y = 2x + 1. On the ASVAB MK section, slope-intercept form (y = mx + b) and point-slope form are commonly tested. Memorize: m = slope, b = y-intercept.

ASVAB Mathematics Knowledge Test 3

Solve for x: 3x² - 12 = 0