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Algebra 2 Practice: Problems, Answers, and Common Mistakes

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A student works through algebra 2 practice problems on paper and checks each answer against the worked solution.
This page gives you 18 Algebra 2 practice problems across six core topics: quadratics and complex numbers, polynomials, rational equations, radical equations, exponential and logarithmic equations, and functions and transformations. Each topic has a Basic, Intermediate, and Challenge problem with worked steps, a final answer, and the most common mistake. Try a problem on your own first, compare the steps, substitute your answer back to check it, and revisit the study guide for any rule you cannot recall.

How to Use This Practice Set

Start with the Basic problem on one topic and solve it before reading the worked answer; after checking, redo only the step that tripped you. Work upward from Basic to Challenge, and return to the Algebra 2 study guide whenever a rule slips your memory. If factoring, fraction arithmetic, or exponent rules keep getting in the way, repair that Algebra 1 gap first with Algebra 1 practice before moving on.
A short practice loop: try the problem on paper, compare the worked steps, then redo the missed step.

Quadratics and Complex Numbers

Basic — Solve x² − 5x + 6 = 0.
  1. Factor: find two numbers that multiply to 6 and add to −5, which are −2 and −3, so (x − 2)(x − 3) = 0.
  1. Set each factor equal to 0: x = 2 or x = 3.
  • Final answer: x = 2 or x = 3.
  • Common error: reversing the signs in the factors and writing x = −2 or x = −3.
Intermediate — Solve x² + 4x + 13 = 0.
  1. Use the quadratic formula with a = 1, b = 4, c = 13: x = (−4 ± √(16 − 52)) / 2.
  1. The discriminant is 16 − 52 = −36, and √(−36) = 6i.
  1. Simplify: x = (−4 ± 6i) / 2 = −2 ± 3i.
  • Final answer: x = −2 + 3i or x = −2 − 3i.
  • Common error: leaving √(−36) unsimplified or forgetting to divide the real part by 2.
Challenge — Solve x² − 6x + 2 = 0 by completing the square.
  1. Move the constant: x² − 6x = −2; half of −6 is −3, and squaring it gives 9, added to both sides.
  1. (x − 3)² = 7, so x − 3 = ±√7.
  1. x = 3 ± √7.
  • Final answer: x = 3 + √7 or x = 3 − √7.
  • Common error: adding 9 to only one side of the equation.
The quadratic equations guide covers factoring, the formula, and the discriminant in more depth.

Polynomials: Factoring, Roots, and Division

Basic — Multiply (x + 3)(x − 5).
  1. Distribute: x² − 5x + 3x − 15.
  1. Combine like terms: x² − 2x − 15.
  • Final answer: x² − 2x − 15.
  • Common error: adding the middle signs incorrectly, such as −5x + 3x = −8x.
Intermediate — Factor 2x² − 5x − 3.
  1. Here a·c = 2·(−3) = −6; find two numbers that multiply to −6 and add to −5, which are −6 and +1.
  1. Split and group: 2x² − 6x + x − 3 = 2x(x − 3) + 1(x − 3).
  1. Factor out (x − 3): (2x + 1)(x − 3).
  • Final answer: (2x + 1)(x − 3).
  • Common error: grouping without factoring the common term from each pair.
Challenge — Factor x³ − 4x² + x + 6 completely.
  1. Test a root with the Rational Zero Theorem: P(2) = 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor.
  1. Divide to obtain x² − 2x − 3, then factor it as (x − 3)(x + 1).
  1. Full factoring: (x − 2)(x − 3)(x + 1).
  • Final answer: (x − 2)(x − 3)(x + 1).
  • Common error: stopping after the first factor and not factoring the quadratic.
The polynomials guide goes deeper into division, the Remainder Theorem, and graphing roots.

Rational Expressions and Equations

Basic — Simplify (x² − 9) / (x² + 5x + 6).
  1. Factor: numerator (x − 3)(x + 3); denominator (x + 2)(x + 3).
  1. Cancel (x + 3) where x ≠ −3, and keep x ≠ −2 as well.
  1. Result: (x − 3) / (x + 2).
  • Final answer: (x − 3) / (x + 2), with x ≠ −3 and x ≠ −2.
  • Common error: cancelling x terms across addition before factoring.
Intermediate — Solve 2/x = 1/(x − 3).
  1. State the domain: x ≠ 0 and x ≠ 3.
  1. Cross-multiply: 2(x − 3) = x, so 2x − 6 = x.
  1. x = 6, which is not excluded.
  • Final answer: x = 6.
  • Common error: clearing denominators without writing the excluded values first.
Challenge — Solve (x + 1)/(x − 1) = 4/(x² − 1).
  1. Domain: x ≠ 1 and x ≠ −1; note x² − 1 = (x − 1)(x + 1).
  1. Multiply by (x − 1)(x + 1): (x + 1)² = 4, so x + 1 = ±2.
  1. Candidates are x = 1 and x = −3; x = 1 is excluded, so reject it and keep x = −3.
  • Final answer: x = −3.
  • Common error: accepting x = 1 even though it makes a denominator zero, which is an extraneous candidate.
The rational expressions guide covers domain, asymptotes, and simplification step by step.

Radical Equations

Basic — Simplify √72.
  1. Factor: 72 = 36 · 2.
  1. √72 = √36 · √2 = 6√2.
  • Final answer: 6√2.
  • Common error: leaving 36 inside the radical or pulling out only part of it.
Intermediate — Solve √(3x + 1) = 4.
  1. Square both sides: 3x + 1 = 16.
  1. 3x = 15, so x = 5; check √16 = 4.
  • Final answer: x = 5.
  • Common error: forgetting to square the right side, 4, as well.
Challenge — Solve √(x + 6) = x.
  1. Square both sides: x + 6 = x², so x² − x − 6 = 0.
  1. Factor: (x − 3)(x + 2) = 0, giving candidates x = 3 and x = −2.
  1. Substitute back: x = 3 gives √9 = 3, which works; x = −2 gives √4 = 2 ≠ −2, so reject it.
  • Final answer: x = 3.
  • Common error: keeping x = −2 without substituting it back, which is an extraneous root.
The radicals guide explains rational exponents and root simplification.

Exponential and Logarithmic Equations

Basic — Evaluate log₃(81).
  1. Ask what power of 3 equals 81: 3⁴ = 81.
  1. log₃(81) = 4.
  • Final answer: 4.
  • Common error: swapping the base and the exponent in the answer.
Intermediate — Solve log₅(x) = 2.
  1. Convert to exponential form: x = 5².
  1. x = 25; check log₅(25) = 2.
  • Final answer: x = 25.
  • Common error: multiplying 5 by 2 instead of raising 5 to the power 2.
Challenge — Solve log₂(x) + log₂(x − 2) = 3.
  1. Domain: x > 2.
  1. Use the product rule: log₂[x(x − 2)] = 3, so x(x − 2) = 2³ = 8.
  1. x² − 2x − 8 = 0 = (x − 4)(x + 2); candidates x = 4 and x = −2, and x = −2 fails x > 2, so reject it.
  • Final answer: x = 4.
  • Common error: dropping the x > 2 domain and keeping x = −2.

Functions and Transformations

Basic — Evaluate f(x) = 2x² − 3 at x = −2.
  1. Substitute: f(−2) = 2(−2)² − 3 = 2(4) − 3.
  1. 8 − 3 = 5.
  • Final answer: f(−2) = 5.
  • Common error: squaring −2 as −4 instead of 4.
Intermediate — Find the domain of f(x) = 1/(x − 4).
  1. The denominator cannot be zero: x − 4 ≠ 0, so x ≠ 4.
  1. All real numbers except 4.
  • Final answer: all real x with x ≠ 4.
  • Common error: writing x > 4 instead of excluding a single point.
Challenge — For f(x) = x², write g(x) = f(x − 3) + 2 and state its vertex.
  1. Substitute: g(x) = (x − 3)² + 2.
  1. A shift written inside f moves the graph right 3, and the +2 moves it up 2.
  1. The vertex (0, 0) of x² moves to (3, 2).
  • Final answer: g(x) = (x − 3)² + 2; vertex (3, 2).
  • Common error: moving left instead of right for a shift written inside f.
The functions guide builds on notation, domain, and graph changes.

Check Your Answers and Sort Your Mistakes

After a set, substitute each answer back into the original equation. Keep the domain you wrote at the start for rational and radical problems, and reject any candidate that falls outside it or fails the substitution. Sort missed problems by type: a calculation slip, a missing domain check, or a misread question, then redo only that kind of problem once more. Whether Algebra 2 feels harder than geometry depends on your background; the study guide compares the two courses if that question is on your mind.
Substitute the answer back into the original equation and reject candidates that fail the check.
The six practice topics above run in the usual dependency order: quadratics and complex numbers, polynomials, rational equations, radical equations, exponential and logarithmic equations, and functions and transformations. Start where you missed the most steps rather than working through every topic in order.
Six practice topics arranged along a light dotted path for choosing where to start.

Practice Algebra 2 with Gauth

Pick the topic where you missed the most steps. The existing Algebra 2 course pairs a unit path with lesson explanations and interactive practice you can use alongside your class materials. If the available units do not match your syllabus, open Gauth Math Courses, select Create Course, and enter one weak topic plus your level to generate a focused interactive micro-course. Redo that topic's Challenge problem afterward.

Sources and Review

OpenStax: Algebra and Trigonometry 2e, accessed October 19, 2026. Standard course scope and formulas; worked problems on this page are solved independently.

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