Nonlinear equations and systems — SAT practice questions

On the SAT Math test, Nonlinear equations and systems covers quadratics, cubics, radical equations, and mixed linear-quadratic systems. A student must solve for a variable, count distinct real solutions, or find a constant that makes two graphs meet exactly once. Questions often present a pair of equations and ask for that constant, or a single equation and ask how many real solutions it has or what the solution is. Choices are usually possible constants, integer counts of roots, or candidate x-values. Traps include keeping an extraneous radical root, missing a real cubic root, and treating one intersection as two.

Section: Math 44 questions on the paper
133 practice questions in the app

Sample questions

Question 1

x^3=4x

How many distinct real solutions does the given equation have?

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Answer: C — 3

Rewriting gives $x^3-4x=0$ , so $x(x-2)(x+2)=0$ . The three distinct real solutions are -2, 0, and 2.

Question 2

y=3x^2+5x-4

y=-37x+k

In the given system of equations, $k$ is a constant. If the graphs of the equations in the given system intersect at exactly one point, what is the value of $k$ ?

A-151
B-42
C7
D151

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Answer: A — -151

Equating the expressions for $y$ gives $3x^2+42x-4-k=0$ . Exactly one intersection means the discriminant is 0: $1764-12(-4-k)=0$ , so $1812+12k=0$ and $k=-151$ .

Question 3

\sqrt{x+4}=x+2

What is the solution to the given equation?

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Answer: C — 0

Squaring both sides gives $x^2+3x=0$ , so the candidates are 0 and -3. Substituting $x=-3$ into the original equation gives $1 \neq -1$ , so $x=-3$ is extraneous. The solution is $x=0$ .

Practice 133 Nonlinear equations and systems questions in the app

Nonlinear equations and systems — SAT practice questions

Sample questions

More SAT topics

Math