Free SAT Math Passport to Advanced Math Practice Test — Questions and Answers

Question 1: Which of the following is equivalent to x² - 9?

(x - 3)²
(x + 3)(x - 3) (Correct answer)
(x - 9)(x + 1)
x(x - 9)

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

x² - 9 is a difference of squares: a² - b² = (a + b)(a - b). So x² - 9 = (x + 3)(x - 3).

Question 2: Solve: x² - 5x + 6 = 0

x = 1 and x = 6
x = 2 and x = 3 (Correct answer)
x = -2 and x = -3
x = -1 and x = 6

Correct answer: x = 2 and x = 3

Factor: (x - 2)(x - 3) = 0. So x = 2 or x = 3.

Question 3: Which is equivalent to (x + 4)²?

x² + 16
x² + 4x + 16
x² + 8x + 16 (Correct answer)
x² - 8x + 16

Correct answer: x² + 8x + 16

(x + 4)² = x² + 2(4)x + 4² = x² + 8x + 16.

Question 4: If f(x) = 3x² - 2x + 1, what is f(2)?

5
7
9 (Correct answer)
13

Correct answer: 9

f(2) = 3(2)² - 2(2) + 1 = 3(4) - 4 + 1 = 12 - 4 + 1 = 9.

Question 5: Which of the following is a zero of f(x) = x² - 2x - 8?

x = -2
x = 2
x = -4
x = 4 (Correct answer)

Correct answer: x = 4

Factor: (x - 4)(x + 2) = 0. So x = 4 or x = -2. Both -2 and 4 are zeros, but only x = 4 appears as a choice here. Answer: D.

Question 6: Simplify: (x³ · x⁴) / x²

x⁵ (Correct answer)
x⁷
x⁹
x¹²

Correct answer: x⁵

Numerator: x³ · x⁴ = x⁷. Then x⁷ / x² = x⁵.

Question 7: What is the vertex of the parabola y = (x - 3)² + 5?

(3, 5) (Correct answer)
(-3, 5)
(3, -5)
(5, 3)

Correct answer: (3, 5)

In vertex form y = (x - h)² + k, the vertex is (h, k). Here h = 3 and k = 5, so the vertex is (3, 5).

Question 8: Which of the following is equivalent to √(x⁴)?

x
x² (Correct answer)
x³
2x

Correct answer: x²

√(x⁴) = x^(4/2) = x².

Question 9: Simplify: (2x²y)(3xy³)

5x³y⁴
6x²y³
6x³y⁴ (Correct answer)
5x²y⁴

Correct answer: 6x³y⁴

Multiply coefficients: 2 × 3 = 6. Add exponents: x^(2+1) = x³, y^(1+3) = y⁴. Result: 6x³y⁴.

Question 10: Which equation represents a parabola that opens downward?

y = x² + 3
y = -2x² + 1 (Correct answer)
y = 2x + 1
y = (x - 1)²

Correct answer: y = -2x² + 1

A parabola opens downward when the coefficient of x² is negative. y = -2x² + 1 has coefficient -2, so it opens downward.

Question 11: If g(x) = 2x + 3, what is g(g(1))?

5
10
13 (Correct answer)
15

Correct answer: 13

g(1) = 2(1) + 3 = 5. g(g(1)) = g(5) = 2(5) + 3 = 13.

Question 12: Solve x² = 25.

x = 5 only
x = -5 only
x = ±5 (Correct answer)
x = ±25

Correct answer: x = ±5

x² = 25 means x = √25 or x = -√25, so x = 5 or x = -5, written as x = ±5.

Question 13: Factor completely: 2x² + 6x

2(x² + 3)
2x(x + 3) (Correct answer)
x(2x + 6)
2(x + 3)²

Correct answer: 2x(x + 3)

The GCF of 2x² and 6x is 2x. Factor out: 2x(x + 3).

Question 14: What is the range of f(x) = x² + 1?

All real numbers
y ≥ 0
y ≥ 1 (Correct answer)
y > 1

Correct answer: y ≥ 1

x² is always ≥ 0, so x² + 1 ≥ 1. The minimum value is 1 (when x = 0), so range is y ≥ 1.

Question 15: Which of the following is equivalent to x^(-2)?

-x²
1/x² (Correct answer)
x²
-1/x²

Correct answer: 1/x²

A negative exponent means reciprocal: x^(-2) = 1/x².

Question 16: The solutions to x² - 4x + 4 = 0 are:

x = 2 only (Correct answer)
x = -2 only
x = 2 and x = -2
x = 4 and x = 1

Correct answer: x = 2 only

x² - 4x + 4 = (x - 2)² = 0. So x = 2 is a repeated root (only one solution).

Question 17: If h(x) = x² - 4 and k(x) = 2x, what is h(k(3))?

14
32 (Correct answer)
36
2

Correct answer: 32

k(3) = 2(3) = 6. h(k(3)) = h(6) = 6² - 4 = 36 - 4 = 32.

Question 18: What is the axis of symmetry of y = x² - 6x + 8?

x = 3 (Correct answer)
x = -3
x = 6
x = -6

Correct answer: x = 3

The axis of symmetry is x = -b/(2a). Here a = 1 and b = -6: x = -(-6)/(2·1) = 6/2 = 3.

Question 19: Which expression equals (x + 3)(x - 3)?

x² + 9
x² - 9 (Correct answer)
x² + 6x - 9
x² - 6x + 9

Correct answer: x² - 9

(x + 3)(x - 3) is the difference of squares pattern: x² - 3² = x² - 9.

Question 20: If 3^x = 81, what is x?

3
4 (Correct answer)
5
27

Correct answer: 4

81 = 3⁴. So 3^x = 3⁴, meaning x = 4.

Question 21: Simplify: (x²)³

x⁵
x⁶ (Correct answer)
3x²
x⁸

Correct answer: x⁶

Power of a power rule: (x^m)^n = x^(mn). So (x²)³ = x^(2×3) = x⁶.

Question 22: Which of the following is a polynomial expression?

3/x + 2
√x + 1
2x³ - 5x + 7 (Correct answer)
x^(-1) + 4

Correct answer: 2x³ - 5x + 7

A polynomial has non-negative integer exponents. Only 2x³ - 5x + 7 meets this definition.

Question 23: What is the discriminant of x² - 4x + 5 = 0?

-4 (Correct answer)
0
4
36

Correct answer: -4

Discriminant = b² - 4ac = (-4)² - 4(1)(5) = 16 - 20 = -4.

Question 24: How many real solutions does x² + 4 = 0 have?

0 (Correct answer)
1
2
4

Correct answer: 0

x² = -4 has no real solutions because x² is always non-negative. The discriminant is negative, confirming no real roots.

Question 25: Which is equivalent to (4x²)/(2x)?

2
2x (Correct answer)
4x
8x

Correct answer: 2x

Simplify: (4x²)/(2x) = (4/2) · (x²/x) = 2x.

Question 26: If p(x) = x³ - x and q(x) = x, what is p(x)/q(x) for x ≠ 0?

x - 1
x² - 1 (Correct answer)
x²
x - x

Correct answer: x² - 1

p(x)/q(x) = (x³ - x)/x = x² - 1, for x ≠ 0.

Question 27: What is the y-intercept of f(x) = 2x² - 3x + 1?

-3
0
1 (Correct answer)
2

Correct answer: 1

The y-intercept is f(0) = 2(0)² - 3(0) + 1 = 1.

Question 28: Solve x² - 7x + 12 = 0 by factoring.

x = 3 and x = 4 (Correct answer)
x = -3 and x = -4
x = 2 and x = 6
x = 1 and x = 12

Correct answer: x = 3 and x = 4

Factor: (x - 3)(x - 4) = 0. So x = 3 or x = 4.

Question 29: Which of the following is equivalent to (a³b²)²?

a⁵b⁴
a⁶b⁴ (Correct answer)
a⁶b²
a⁹b⁴

Correct answer: a⁶b⁴

Apply power rule to each factor: (a³)² = a⁶ and (b²)² = b⁴. Result: a⁶b⁴.

Question 30: If f(x) = x² - 1, what is f(-3)?

-10
7
8 (Correct answer)
10

Correct answer: 8

f(-3) = (-3)² - 1 = 9 - 1 = 8.

Question 31: An equation has two real roots that are both positive. Which condition must hold?

Discriminant = 0
Discriminant < 0
Discriminant > 0 and sum/product of roots are positive (Correct answer)
The coefficient of x² is negative

Correct answer: Discriminant > 0 and sum/product of roots are positive

For two real roots: discriminant > 0. For both positive: sum = -b/a > 0 and product = c/a > 0.

Question 32: Simplify: √(50)

5
5√2 (Correct answer)
10√5
25√2

Correct answer: 5√2

√50 = √(25 × 2) = √25 × √2 = 5√2.

Question 33: What is the domain of f(x) = 1/(x - 4)?

All real numbers
x ≠ 0
x ≠ 4 (Correct answer)
x > 4

Correct answer: x ≠ 4

The function is undefined when the denominator equals zero: x - 4 = 0 → x = 4. So domain is all real numbers except x = 4.

Question 34: Which of the following equations has exactly one real root?

x² + 4x + 4 = 0 (Correct answer)
x² + 4x + 3 = 0
x² + 4x + 5 = 0
x² - 4 = 0

Correct answer: x² + 4x + 4 = 0

x² + 4x + 4 = (x + 2)² = 0, giving x = -2 as a repeated root — exactly one real root. Discriminant = 0 confirms this.

Question 35: If 2^(x+1) = 16, what is x?

2
3 (Correct answer)
4
7

Correct answer: 3

16 = 2⁴. So 2^(x+1) = 2⁴ → x + 1 = 4 → x = 3.

Question 36: A function f is defined by f(x) = x² - 6x + 9. Which of the following is true?

It factors as (x-3)² and its vertex is (3,0) (Correct answer)
It has two distinct real roots
Its minimum value is 9
It opens downward

Correct answer: It factors as (x-3)² and its vertex is (3,0)

f(x) = (x - 3)². The vertex is at (3, 0) and it opens upward. It has exactly one (repeated) root at x = 3.

Free SAT Math Passport to Advanced Math Practice Test

Which of the following is equivalent to x² - 9?