Practice Math — Advanced Math questions for the Scholastic Assessment Test. Every question includes a full explanation of why the correct answer is right and why the tempting distractors are wrong.
10 example questions with full explanations. Use the interactive practice above to work through the complete set.
Question 1medium
The function f(x) = x² - 6x + 8 is given. Select all that apply.
- A.The x-intercepts of f are x = 2 and x = 4.✓
- B.The vertex of f is at the point (3, -1).✓
- C.The y-intercept of f is at (0, -8).
- D.The parabola opens upward.✓
- E.The axis of symmetry is x = 4.
Correct answer: A, B, D
Factoring f(x) = (x - 2)(x - 4) gives x-intercepts at x = 2 and x = 4, so A is correct. The vertex x-coordinate is -(-6)/(2·1) = 3, and f(3) = 9 - 18 + 8 = -1, giving vertex (3, -1), so B is correct. The leading coefficient is positive (1), so the parabola opens upward, making D correct. C is wrong because f(0) = 8, not -8. E is wrong because the axis of symmetry is x = 3, not x = 4.
Question 2medium
Which of the following expressions are equivalent to (2x² + 5x - 3) / (x + 3)? Select all that apply.
- A.2x - 1, for x ≠ -3✓
- B.2x + 1, for x ≠ -3
- C.(2x - 1)(x + 3) / (x + 3), for x ≠ -3✓
- D.2 - (1/x), for x ≠ 0 and x ≠ -3
Correct answer: A, C
Factoring the numerator: 2x² + 5x - 3 = (2x - 1)(x + 3). Dividing by (x + 3) for x ≠ -3 gives 2x - 1, so A is correct. C is equivalent because it simply shows the factored form before cancellation, which also simplifies to 2x - 1 for x ≠ -3. B is incorrect because 2x + 1 does not equal 2x - 1. D simplifies to (2x - 1)/x, which is not equal to 2x - 1 in general.
Question 3medium
A population of bacteria is modeled by P(t) = 500 · (2)^(t/3), where t is in hours. Select all that apply.
- A.The initial population at t = 0 is 500.✓
- B.The population doubles every 3 hours.✓
- C.At t = 6, the population is 2,000.✓
- D.The growth factor per hour is 2.
Correct answer: A, B, C
At t = 0, P(0) = 500 · 2^0 = 500, so A is correct. The exponent t/3 means the quantity doubles every time t increases by 3, so B is correct. At t = 6, P(6) = 500 · 2^(6/3) = 500 · 4 = 2000, so C is correct. D is incorrect: the growth factor per hour is 2^(1/3) ≈ 1.26, not 2. The factor of 2 applies every 3 hours, not every hour.
Question 4medium
The quadratic equation 2x² - 8x + 6 = 0 is given. Select all that apply.
- A.The solutions are x = 1 and x = 3.✓
- B.The equation can be factored as 2(x - 1)(x - 3) = 0.✓
- C.Completing the square gives (x - 2)² = 2.
- D.The sum of the solutions is 4.✓
- E.The product of the solutions is 6.
Correct answer: A, B, D
Dividing by 2 gives x² - 4x + 3 = 0, factoring as (x - 1)(x - 3) = 0, yielding x = 1 and x = 3. So A and B are correct. For D, the sum of the roots is 1 + 3 = 4, confirmed also by -b/a = 8/2 = 4. Correct. For C, completing the square: x² - 4x + 3 = 0 → (x - 2)² = 1, not 2, so C is incorrect. For E, the product of the roots is 1 × 3 = 3, not 6 (c/a = 6/2 = 3), so E is incorrect.
Question 5easy
Let f(x) = 3x² − 12x + 9. Select all that apply.
- A.f(0) = 9✓
- B.f(1) = 0✓
- C.f(3) = 0✓
- D.f(2) = 3
- E.f(−1) = 9
Correct answer: A, B, C
Evaluating each: f(0) = 3(0)² − 12(0) + 9 = 9 ✓; f(1) = 3(1) − 12(1) + 9 = 3 − 12 + 9 = 0 ✓; f(3) = 3(9) − 12(3) + 9 = 27 − 36 + 9 = 0 ✓; f(2) = 3(4) − 12(2) + 9 = 12 − 24 + 9 = −3, not 3 ✗; f(−1) = 3(1) − 12(−1) + 9 = 3 + 12 + 9 = 24, not 9 ✗. Options A, B, and C are correct. This question tests direct function evaluation using substitution.
Question 6easy
Which of the following expressions are equivalent to (2x³)(3x⁴)? Select all that apply.
- A.6x⁷✓
- B.6x¹²
- C.2(3x⁷)✓
- D.x⁷ · 6✓
Correct answer: A, C, D
When multiplying powers with the same base, multiply the coefficients and add the exponents: (2x³)(3x⁴) = (2·3)x^(3+4) = 6x⁷. Option A is 6x⁷ ✓; Option B is 6x¹², which incorrectly multiplies the exponents instead of adding them ✗; Option C is 2(3x⁷) = 6x⁷ ✓; Option D is x⁷ · 6 = 6x⁷ ✓. A common error is multiplying exponents (giving x¹²) rather than adding them.
Question 7easy
The polynomial p(x) = x² + 5x + 6 can be factored. Select all that apply.
- A.p(x) = (x + 2)(x + 3)✓
- B.p(x) has a root at x = −2✓
- C.p(x) has a root at x = 2
- D.p(x) has a root at x = −3✓
Correct answer: A, B, D
Factoring x² + 5x + 6: we need two numbers that multiply to 6 and add to 5, which are 2 and 3. So p(x) = (x + 2)(x + 3) ✓ (Option A). Setting each factor to zero gives roots x = −2 ✓ (Option B) and x = −3 ✓ (Option D). Option C claims a root at x = 2: p(2) = 4 + 10 + 6 = 20 ≠ 0, so C is incorrect. A common distractor error is confusing the sign of the roots from the factored form.
Question 8easy
The function f(x) = 2(x − 3)² + 5 is written in vertex form. Select all that apply.
- A.The vertex of the parabola is (3, 5)✓
- B.The vertex of the parabola is (−3, 5)
- C.The parabola opens upward✓
- D.The y-intercept is 5
- E.The minimum value of f(x) is 5✓
Correct answer: A, C, E
In vertex form f(x) = a(x − h)² + k, the vertex is (h, k). Here h = 3 and k = 5, so the vertex is (3, 5) ✓ (A). Since a = 2 > 0, the parabola opens upward ✓ (C), and the minimum value is k = 5 ✓ (E). Option B incorrectly uses −3 as the x-coordinate of the vertex, a common sign error. Option D is wrong: the y-intercept is f(0) = 2(0−3)² + 5 = 2(9) + 5 = 23, not 5.
Question 9easy
Which of the following is equivalent to the expression (3x²y)(4xy³)?
- A.7x³y⁴
- B.12x²y³
- C.12x³y⁴✓
- D.7x²y³
Correct answer: C
To multiply two monomials, multiply the coefficients and add the exponents of like bases. The coefficients are 3 and 4, giving 3 × 4 = 12. For x: x² × x¹ = x^(2+1) = x³. For y: y¹ × y³ = y^(1+3) = y⁴. So the result is 12x³y⁴. Choice A incorrectly adds the coefficients (3 + 4 = 7) instead of multiplying them. Choices B and D fail to correctly apply the rule of adding exponents.
Question 10easy
What is the simplified form of the expression √(49x⁴)?
Correct answer: D
To simplify √(49x⁴), take the square root of each factor separately. √49 = 7, and √(x⁴) = x^(4/2) = x². Therefore, √(49x⁴) = 7x². Choice A incorrectly computes √(x⁴) = x instead of x². Choice B removes the square root from the coefficient but not the variable. Choice C takes the square root of 49 correctly but leaves the exponent on x unchanged.