Algebra 2 Radical and Rational Equations Test Solutions and Scoring (PDF)

Algebra 2 math test solutions: Radical and Rational Equations – Maths PDF

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

Use this answer key and scoring guide to grade the test on radical and rational equations (A-REI.A.2). Each solution shows the full reasoning, and the scoring note explains how many points to award for correct setup, correct work, and a correct final answer. Partial credit rewards a sound method even when a small arithmetic slip changes the result.

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

Radical rational equations test answers: worked solutions

Question 1 – Basic radical equation [1 pt]

\(3x + 1 = 49\), so \(x = 16\).

\(x = 16\)

Scoring: 1 point for the correct solution with a check.

Question 2 – Extraneous solution [3 pts]

Only x = 4 is a solution.

\(x + 12 = x^{2}\), so \(x^{2} – x – 12 = 0\) and \((x – 4)(x + 3) = 0\).

Check \(x = 4\): \(\sqrt{16} = 4\). Check \(x = -3\): \(\sqrt{9} = 3 \ne -3\).

\(x = 4\); \(x = -3\) is extraneous

Scoring: 1 point for squaring and solving, 1 point for checking, 1 point for identifying the extraneous solution.

Question 3 – Rational equation [2 pts]

Cross multiply: \(5x = 2(x + 1)\), so \(3x = 2\).

\(x = \frac{2}{3}\)

Scoring: 1 point for clearing fractions, 1 point for the solution.

Question 4 – No solution [3 pts]

Multiply by \(x + 3\): \(x = -3 + 2(x + 3)\), so \(x = 2x + 3\) and \(x = -3\).

But \(x = -3\) makes the denominator 0.

No solution; x = -3 is extraneous

Scoring: 1 point for clearing fractions, 1 point for solving, 1 point for rejecting x = -3.

Question 5 – Binomial equals radical [2 pts]

\(x^{2} – 6x + 9 = x – 1\), so \(x^{2} – 7x + 10 = 0\) and \((x – 2)(x – 5) = 0\).

Check \(x = 5\): \(2 = 2\). Check \(x = 2\): \(-1 = 1\), false.

\(x = 5\)

Scoring: 1 point for squaring and solving, 1 point for checking.

Question 6 – Two machines [3 pts]

\(\frac{1}{x} + \frac{1}{x + 6} = \frac{1}{4}\). Multiply by \(4x(x + 6)\): \(4(x + 6) + 4x = x(x + 6)\).

\(8x + 24 = x^{2} + 6x\), so \(x^{2} – 2x – 24 = 0\) and \((x – 6)(x + 4) = 0\).

1/6 + 1/12 = 1/4.

6 hours

Scoring: 1 point for the equation, 1 point for the quadratic, 1 point for choosing the positive solution.

Question 7 – Free fall [2 pts]

\(\sqrt{h} = 20\), so \(h = 400\).

400 feet

Scoring: 1 point for isolating the root, 1 point for the answer.

Question 8 – Cube root [2 pts]

Cube both sides: \(x + 2 = 27\).

\(x = 25\)

Scoring: 1 point for cubing both sides, 1 point for the solution.

Question 9 – Check by substitution [2 pts]

A solution makes both columns equal: only \(x = 5\) gives \(3 = 3\).

x = 5

Scoring: 1 point for comparing the columns, 1 point for the answer.

Total score

Total

20 points

Key facts behind this radical rational equations test answers

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 test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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