Nonlinear equations and systems — SAT practice questions

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Practice 139 Nonlinear equations and systems questions in the app

What this topic is

Nonlinear equations and systems include quadratics, equations with a square root or absolute value, cubics, and systems pairing a line with a parabola.

Solving an equation with a radical means squaring both sides, which can create extraneous solutions, so every result must be checked in the original equation. When a polynomial equals a multiple of x, dividing both sides by x throws away the solution x equals zero; moving everything to one side and factoring keeps it.

For a line and a parabola that meet at exactly one point, setting them equal gives a quadratic whose discriminant must be zero. Questions ask for solutions, for how many there are.

Sample questions

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Question 1Mid

x^{3} = 4 x

How many distinct real solutions does the given equation have?

1✗ Not this one2✗ Not this one3✓ Correct4✗ Not this one

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Answer: C — 3

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

Question 2Harder

y = 3 x^{2} + 5 x - 4

y = - 37 x + 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 $k$ ?

- 151

✓ Correct

- 42

✗ Not this one7✗ Not this one151✗ Not this one

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Answer: A — $- 151$

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

Question 3Easier

\sqrt{x + 4} = x + 2

What is the solution to the given equation?

- 4

✗ Not this one

- 3

✗ Not this one0✓ Correct2✗ Not this one

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Answer: C — 0

Squaring both sides gives $x^{2} + 3 x = 0$ , 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 $x = 0$ .

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Practice 139 Nonlinear equations and systems questions in the app

Nonlinear equations and systems — SAT practice questions

What this topic is

Sample questions

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Math