Algebra 2 Radical and Rational Equations Worksheet (PDF)

Algebra 2 math worksheet: Radical and Rational Equations – Maths PDF

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This radical rational equations worksheet for Algebra 2 covers radical and rational equations (Common Core A-REI.A.2), with a free PDF to print.

These 20 problems practice radical and rational equations (Common Core A-REI.A.2). The first problems check isolating a radical or clearing a denominator, the middle problems add checking for extraneous solutions, and the last six are word problems and multi-step tasks that need work, distance, and free-fall applications.

Work in order and show every step. Write each answer in simplest form, and open the answer key only after you have tried the problem yourself.

Algebra 2 · Common Core State Standards: A-REI.A.2

Radical rational equations worksheet: exercises 1 to 10

Exercise 1 – Basic radical equation

Solve \(\sqrt{x + 5} = 4\).

Exercise 2 – Radical with a coefficient

Solve \(\sqrt{2x – 1} = 5\).

Exercise 3 – Extraneous solution

Graphs of both sides.

Solve \(\sqrt{x + 6} = x\). The graphs of both sides are shown.

Exercise 4 – Radical equal to a binomial

Solve \(x – 1 = \sqrt{x + 11}\).

Exercise 5 – Both solutions valid

Graphs of both sides.

Solve \(\sqrt{3x + 4} – 2 = x\) and check each answer.

Exercise 6 – Cube root equation

Solve \(\sqrt[3]{x – 3} = 2\).

Exercise 7 – Radicals on both sides

Solve \(\sqrt{x + 7} = \sqrt{2x – 1}\).

Exercise 8 – Two radicals

Solve \(\sqrt{x + 3} + \sqrt{x} = 3\).

Exercise 9 – Proportion

Solve \(\frac{6}{x} = \frac{3}{4}\).

Exercise 10 – Rational equation

Solve \(\frac{3}{x + 2} = \frac{1}{x}\).

Exercises 11 and up

Exercise 11 – Clear the fractions

Solve \(\frac{2}{x} + \frac{1}{3} = 1\).

Exercise 12 – An extraneous solution

Steps and check.

Solve \(\frac{x}{x – 2} = \frac{2}{x – 2} + 3\). Decide if the candidate is valid.

Exercise 13 – Common denominator

Solve \(\frac{1}{x – 1} + \frac{1}{x + 1} = \frac{4}{x^{2} – 1}\).

Exercise 14 – Both candidates excluded

Solve \(\frac{x}{x + 3} + \frac{3}{x – 3} = \frac{18}{x^{2} – 9}\).

Exercise 15 – Rational leads to a quadratic

Solve \(\frac{6}{x} = x – 1\).

Exercise 16 – Filling a pool

Fill rates per hour.

Hose A fills Liam’s pool \(3\) hours faster than hose B, and together they fill it in 2 hours. Write an equation for the time \(t\) that hose A needs alone, and find how long each hose takes alone.

Exercise 17 – Boat and current

A river trip.

A boat travels 24 miles upstream and 24 miles back downstream in 5 hours in a river with a 2 mph current. Find the boat’s speed in still water. Use \(\frac{24}{v – 2} + \frac{24}{v + 2} = 5\).

Exercise 18 – Free fall

The time in seconds for a ball to fall \(h\) feet is \(t = \frac{\sqrt{h}}{4}\). Maya drops a ball and it takes 3 seconds to land. From what height was it dropped?

Exercise 19 – Skid marks

A car’s speed in mph from skid marks of \(d\) feet is \(s = \sqrt{24d}\). A car skidded and its speed was estimated at 36 mph. How long was the skid?

Exercise 20 – Find the error

Noah solved \(\sqrt{x + 6} = x\) and wrote \(x = 3\) or \(x = -2\). He did not check. Explain what went wrong.

Key facts behind this radical rational equations worksheet

Key facts

Isolate the radical before squaring.
Squaring can create extraneous solutions; always check.
Rational equations: multiply by the LCD to clear fractions.
Reject answers that make a denominator 0.
A principal square root is never negative.

How to use this radical rational equations worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

All Algebra 2 chapters: Algebra 2 math.

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