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SAT Advanced Math Quizzes

Practice 10 SAT Advanced Math questions involving equivalent expressions, quadratics, absolute value, radicals, exponentials, polynomials, rational expressions, nonlinear equations, nonlinear systems, and nonlinear functions. Use hints when needed and review the explanation after each answer.

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1 Question 1 of 10
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Which expression is equivalent to x² + 9x + 20?

Hint: Find two numbers whose product is 20 and whose sum is 9.
Explanation: The numbers 4 and 5 multiply to 20 and add to 9. Therefore x² + 9x + 20 = (x + 4)(x + 5).
2 Question 2 of 10
Not answered

For x ≠ 4, which expression is equivalent to (x² - 16) / (x - 4)?

Hint: Factor the numerator as a difference of squares.
Explanation: x² - 16 = (x - 4)(x + 4). For x ≠ 4, the factor x - 4 cancels, leaving x + 4.
3 Question 3 of 10
Not answered

What is the greater solution of x² - 11x + 24 = 0?

Hint: Factor the quadratic using two numbers that multiply to 24 and add to -11.
Explanation: x² - 11x + 24 = (x - 3)(x - 8). The solutions are x = 3 and x = 8, so the greater solution is 8.
4 Question 4 of 10
Not answered

The equation |2x - 5| = 9 has two solutions. What is the sum of the two solutions?

Hint: Solve 2x - 5 = 9 and 2x - 5 = -9 separately.
Explanation: From 2x - 5 = 9, x = 7. From 2x - 5 = -9, x = -2. Their sum is 7 + (-2) = 5.
5 Question 5 of 10
Not answered

What is the solution of √(x + 5) = x - 1?

Hint: Because the square root is nonnegative, x - 1 must be nonnegative. Square both sides, solve, then check the candidates in the original equation.
Explanation: Squaring gives x + 5 = (x - 1)² = x² - 2x + 1, so x² - 3x - 4 = 0. This factors as (x - 4)(x + 1) = 0, giving 4 and -1. Only x = 4 satisfies the original equation, since the right side must be nonnegative.
6 Question 6 of 10
Not answered

A population is modeled by P(t) = 600(1.08)^t, where t is the number of years after the start of the study. What does 1.08 represent?

Hint: An exponential growth factor has the form 1 + r, where r is the growth rate as a decimal.
Explanation: The factor 1.08 equals 1 + 0.08, so the model represents an 8% increase each year.
7 Question 7 of 10
Not answered

The function g(x) = (x - 3)² - 7 is graphed in the xy-plane. What is the minimum value of g(x)?

Hint: The function is in vertex form (x - h)² + k.
Explanation: The vertex is (3, -7). Since the squared term has a positive coefficient, the parabola opens upward, so the minimum value of g(x) is -7.
8 Question 8 of 10
Not answered

The polynomial p(x) = x³ - 4x² - x + 4 has how many positive real zeros?

Hint: Factor by grouping: group the first two terms and the last two terms.
Explanation: p(x) = x²(x - 4) - 1(x - 4) = (x - 4)(x² - 1) = (x - 4)(x - 1)(x + 1). The real zeros are 4, 1, and -1. Two of them are positive.
9 Question 9 of 10
Not answered

The graphs of y = x² and y = 4x - 3 intersect at two points. What is the sum of the x-coordinates of the intersection points?

Hint: Set the two expressions for y equal and solve the resulting quadratic.
Explanation: Set x² = 4x - 3. Then x² - 4x + 3 = 0, which factors as (x - 1)(x - 3) = 0. The x-coordinates are 1 and 3, and their sum is 4.
10 Question 10 of 10
Not answered

An exponential function h is given by h(x) = ab^x, where a and b are positive constants. If h(0) = 5 and h(2) = 45, what is h(3)?

Hint: Use h(0) to find a. Then use h(2) to find b.
Explanation: Since h(0) = a = 5, we have 5b² = 45, so b² = 9. Because b is positive, b = 3. Therefore h(3) = 5(3³) = 5 × 27 = 135.

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SAT Advanced Math map

Advanced Math is where SAT algebra becomes nonlinear and structure becomes the shortcut

These 10 questions practice equivalent expressions, quadratics, absolute value, radicals, exponentials, polynomial factoring, nonlinear systems, and nonlinear functions. The goal is not simply to calculate more; it is to recognize which algebraic form exposes roots, growth, domain restrictions, or transformations most clearly.

Equivalent expressions
Nonlinear equations & systems
Nonlinear functions
Equivalent-expression lens

Different forms reveal different features

The first questions reward factoring before computing.

x² + 9x + 20
(x + 4)(x + 5)
Expanded form shows coefficients; factored form immediately shows the structure of the factors and is often better for solving.
Quadratic structure

Roots and vertex answer different questions about a parabola

For x² − 11x + 24 = 0, factoring exposes the x-intercepts immediately.

x² − 11x + 24 = 0
(x − 3)(x − 8) = 0
x = 3 or x = 8
Nonlinear-equation filter

Squaring can create candidate solutions that were never valid in the original equation

Generate candidates

√(x + 5) = x − 1

Squaring leads to candidates x = 4 and x = −1.

Check the original

x = 4 works
x = −1 fails because the right side is negative
Exponential growth

The base of an exponential model is a repeated multiplier

In a model such as P(t) = 600(1.08)^t, the 1.08 factor means each new value is 108% of the previous value — an 8% increase per period.

Start100%
Next period× growth factor
Next again× same factor
Patternmultiplicative
Polynomial factor tree

Factoring a polynomial turns a complicated expression into a list of possible zeros

The polynomial question can be handled by grouping and then factoring a difference of squares.

x³ − 4x² − x + 4
(x − 4)(x² − 1)
(x − 4)(x − 1)(x + 1)
Nonlinear system

An intersection problem becomes an equation when both expressions represent y

For y = x² and y = 4x − 3, set the two right sides equal.

x² = 4x − 3
x² − 4x + 3 = 0
(x − 1)(x − 3) = 0
x-values: 1 and 3
Advanced Math traps

Most mistakes come from losing structure or forgetting restrictions

Canceling factorsRemember excluded values in rational expressions.
Extraneous rootsCheck radical-equation candidates in the original equation.
Function formUse vertex, factored, or exponential form for the feature being asked.
Nonlinear systemsEquate expressions before trying to read intersection values.
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