Question 1Easier
What is the solution of ?
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Answer: C — 4
Subtracting from both sides gives , so . The value 0 is excluded because it makes a denominator zero.
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Mathematics
Radical and rational equations — ACT practice questions
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What this topic is
Radical and rational equations carry a special risk: solving them can produce answers that do not actually work.
Squaring both sides of an equation with a square root may introduce an extraneous solution, so every result needs to be checked in the original equation. Rational equations, which have variables in denominators, are usually cleared by multiplying through by a common denominator, but any value that makes a denominator zero must be rejected.
Work-rate problems, where two machines together finish a job, lead to rational equations because their rates, the reciprocals of their times, add. ACT Mathematics answer choices often include the extraneous value as a distractor.
Sample questions
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Question 2Mid
One pump fills a tank in hours, and another pump fills the same tank in hours. Working together, the two pumps fill the tank in 4 hours. What is the value of ?
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Answer: B — 6
Clearing denominators gives , so the candidates are 6 and . Because a completion time is positive, .
Question 3Mid
What is the solution of ?
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Answer: C —
Squaring gives , whose roots are . The smaller root makes negative, so only works.
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More ACT topics
Mathematics
- Real number system and number properties
- Integer and rational exponents
- Radicals and roots
- Complex numbers
- Vectors and matrices
- Fractions, decimals, and operations
- Percents and percent change
- Ratios, rates, and proportional relationships
- Linear equations and expressions in one variable
- Inequalities (linear and compound)
- Systems of equations
- Polynomial expressions and factoring
- Quadratic equations
- Exponential relationships
- Function notation and evaluating functions
- Linear functions and graphs
- Quadratic and polynomial functions
- Radical and rational functions
- Piecewise functions
- Exponential and logarithmic functions
- Key features and transformations of graphs
- Lines, angles, and polygons
- Triangles — congruence, similarity, Pythagorean theorem
- Circles — arcs, sectors, angles, equations
- Surface area and volume of solids
- Coordinate geometry — distance, midpoint, slope, line equations
- Conic sections
- Right-triangle trigonometry and trig functions
- Measures of center and spread
- Data collection methods and study design
- Bivariate data and relationships (scatterplots, correlation)
- Probability and sample spaces
- Counting, combinations, and permutations
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