Radical and rational equations — ACT practice questions

Radical and rational equations on the ACT Mathematics test involve solving for a variable under a square root or in a denominator, and translating combined-work stories into rate equations. A student must isolate a radical and square both sides, or multiply through by a common denominator, then check every candidate so extraneous roots and zeros of denominators are discarded. Items usually ask for the solution of a displayed equation or for an unknown time when two machines work together. Choices tend to be single numerical values, and common traps are keeping an extraneous root after squaring or an x that makes a fraction undefined.

Section: Mathematics 45 questions on the paper
38 practice questions in the app

Sample questions

Question 1

What is the solution of $\dfrac{3}{x}+\dfrac{1}{2}=\dfrac{5}{x}$ ?

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Answer: C — 4

Subtracting $\dfrac{3}{x}$ from both sides gives $\dfrac{2}{x}=\dfrac{1}{2}$ , so $x=4$ . The value 0 is excluded because it makes a denominator zero.

Question 2

One pump fills a tank in $x$ hours, and another pump fills the same tank in $x+6$ hours. Working together, the two pumps fill the tank in 4 hours. What is the value of $x$ ?

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Answer: B — 6

Clearing denominators gives $x^2-2x-24=0$ , so the candidates are 6 and -4. Because a completion time is positive, $x=6$ .

Question 3

What is the solution of $\sqrt{2x+15}=x+3$ ?

A $-2-\sqrt{10}$
B $2-\sqrt{10}$
C $-2+\sqrt{10}$
D $2+\sqrt{10}$

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Answer: C — $-2+\sqrt{10}$

Squaring gives $x^2+4x-6=0$ , whose roots are $-2\pm\sqrt{10}$ . The smaller root makes $x+3$ negative, so only $-2+\sqrt{10}$ works.

Practice 38 Radical and rational equations questions in the app

Radical and rational equations — ACT practice questions

Sample questions

More ACT topics

Mathematics