• 108 questions

Algebra II Practice Test

An unofficial Algebra 2 quiz in seven units, from complex numbers to logarithms and sequences, each answer worked out.

Simplify (2 − i)².

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Unofficial practice questions, not affiliated with any school, district or state education agency. Bank checked October 9, 2026.

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All 108 questions

Quadratics and complex numbers 18

  • At which points do the graphs of y = x² − 1 and y = x + 1 meet? Common Core State Standards for Mathematics, HSA.REI.C.7: solve a simple system consisting of a linear equation and a quadratic equation in two variables
  • How many different real solutions does x² − 6x + 9 = 0 have? Common Core State Standards for Mathematics, HSA.REI.B.4.b: solve quadratic equations by taking square roots, completing the square, the quadratic formula and factoring, writing complex solutions as a ± bi
  • Simplify (2 − i)². Common Core State Standards for Mathematics, HSN.CN.A.2: use i² = −1 and the properties of operations to add, subtract and multiply complex numbers
  • Simplify (3 + 2i) + (5 − 7i). Common Core State Standards for Mathematics, HSN.CN.A.2: use i² = −1 and the properties of operations to add, subtract and multiply complex numbers
  • Simplify (3 + 2i)(1 − 4i). Common Core State Standards for Mathematics, HSN.CN.A.2: use i² = −1 and the properties of operations to add, subtract and multiply complex numbers
  • Simplify (5 + 2i)(5 − 2i). Common Core State Standards for Mathematics, HSN.CN.A.2: use i² = −1 and the properties of operations to add, subtract and multiply complex numbers
Show all 108 questions (102 more)

Quadratics and complex numbers, continued

  • Simplify √(−36). Common Core State Standards for Mathematics, HSN.CN.A.1: there is a complex number i with i² = −1, and every complex number has the form a + bi
  • Simplify √(−50). Common Core State Standards for Mathematics, HSN.CN.A.1: there is a complex number i with i² = −1, and every complex number has the form a + bi
  • Solve 3x² + 5x − 2 = 0. Common Core State Standards for Mathematics, HSA.REI.B.4.b: solve quadratic equations by taking square roots, completing the square, the quadratic formula and factoring, writing complex solutions as a ± bi
  • Solve x² + 2x + 5 = 0. Common Core State Standards for Mathematics, HSN.CN.C.7: solve quadratic equations with real coefficients that have complex solutions
  • Solve x² + 25 = 0. Common Core State Standards for Mathematics, HSN.CN.C.7: solve quadratic equations with real coefficients that have complex solutions
  • Solve x² − 2x − 4 = 0. Common Core State Standards for Mathematics, HSA.REI.B.4.b: solve quadratic equations by taking square roots, completing the square, the quadratic formula and factoring, writing complex solutions as a ± bi
  • Solve x² − 4x + 13 = 0. Common Core State Standards for Mathematics, HSN.CN.C.7: solve quadratic equations with real coefficients that have complex solutions
  • What is i¹⁰? Common Core State Standards for Mathematics, HSN.CN.A.2: use i² = −1 and the properties of operations to add, subtract and multiply complex numbers
  • What is the discriminant, b² − 4ac, of 2x² − 3x + 5 = 0? Common Core State Standards for Mathematics, HSA.REI.B.4.b: solve quadratic equations by taking square roots, completing the square, the quadratic formula and factoring, writing complex solutions as a ± bi
  • What is the maximum value of f(x) = −x² + 6x − 5? Common Core State Standards for Mathematics, HSA.SSE.B.3.b: complete the square in a quadratic expression to reveal the maximum or minimum value
  • What is the vertex of f(x) = −2(x − 3)² + 7? Common Core State Standards for Mathematics, HSF.IF.C.8.a: use factoring and completing the square in a quadratic function to show zeros, extreme values and symmetry of the graph
  • Write y = x² + 8x + 10 in vertex form. Common Core State Standards for Mathematics, HSA.REI.B.4.a: use completing the square to transform a quadratic equation into the form (x − p)² = q

Polynomial functions and zeros 16

  • Counting repeated and complex zeros, how many zeros does a polynomial of degree 5 have? Common Core State Standards for Mathematics, HSN.CN.C.9: know the Fundamental Theorem of Algebra
  • Divide 2x² + 7x + 9 by x + 2. Common Core State Standards for Mathematics, HSA.APR.D.6: rewrite simple rational expressions in different forms
  • Divide x³ + 2x² − 5x − 6 by x − 2. Common Core State Standards for Mathematics, HSA.APR.D.6: rewrite simple rational expressions in different forms
  • Expand (x + 3)³. Common Core State Standards for Mathematics, HSA.APR.A.1: add, subtract and multiply polynomials
  • Factor x³ + 2x² − 9x − 18. Common Core State Standards for Mathematics, HSA.SSE.A.2: use the structure of an expression to identify ways to rewrite it
  • Factor x³ + 64. Common Core State Standards for Mathematics, HSA.SSE.A.2: use the structure of an expression to identify ways to rewrite it
  • Factor x⁴ − 16 completely. Common Core State Standards for Mathematics, HSA.SSE.A.2: use the structure of an expression to identify ways to rewrite it
  • For which value of k is x − 3 a factor of x³ − 2x² − 5x + k? Common Core State Standards for Mathematics, HSA.APR.B.2: know and apply the Remainder Theorem
  • How does the graph of f(x) = −2x⁴ + 3x² + 1 behave at its two ends? Common Core State Standards for Mathematics, HSF.IF.C.7.c: graph polynomial functions, identifying zeros and showing end behavior
  • Multiply (x + 2)(x² − 3x + 4). Common Core State Standards for Mathematics, HSA.APR.A.1: add, subtract and multiply polynomials
  • The polynomial f(x) = (x − 1)²(x + 3) has a zero at x = 1. What does its graph do there? Common Core State Standards for Mathematics, HSF.IF.C.7.c: graph polynomial functions, identifying zeros and showing end behavior
  • Simplify (a + b)² − (a − b)². Common Core State Standards for Mathematics, HSA.APR.C.4: prove polynomial identities and use them to describe numerical relationships
  • What are the zeros of f(x) = x(x − 4)(x + 5)? Common Core State Standards for Mathematics, HSA.APR.B.3: identify zeros of polynomials when suitable factorizations are available
  • What are the zeros of the cubic function in the graph? (Picture: A coordinate grid from −5 to 5 with an S-shaped curve. It rises from the lower left, crosses the x-axis at −2, peaks above the axis, crosses the y-axis at 3, crosses the x-axis at 1, dips below, and crosses the x-axis at 3 before rising.) Common Core State Standards for Mathematics, HSF.IF.C.7.c: graph polynomial functions, identifying zeros and showing end behavior
  • What is the remainder when x³ − 4x² + 2x + 5 is divided by x − 2? Common Core State Standards for Mathematics, HSA.APR.B.2: know and apply the Remainder Theorem
  • Which polynomial with leading coefficient 1 has the zeros 2, −1 and 3? Common Core State Standards for Mathematics, HSA.APR.B.3: identify zeros of polynomials when suitable factorizations are available

Rational expressions and equations 16

  • Add 2/(3x) + 5/(6x). Common Core State Standards for Mathematics, HSA.APR.D.7: add, subtract, multiply and divide rational expressions
  • Add 4/x + 3/(x − 2). Common Core State Standards for Mathematics, HSA.APR.D.7: add, subtract, multiply and divide rational expressions
  • Divide (x + 1)/6 by (x² − 1)/3. Common Core State Standards for Mathematics, HSA.APR.D.7: add, subtract, multiply and divide rational expressions
  • For which value of x is (x + 4)/(x − 7) undefined? Common Core State Standards for Mathematics, HSF.IF.C.7.d: graph rational functions, identifying zeros and asymptotes
  • Multiply x/(x + 3) by (x² − 9)/x². Common Core State Standards for Mathematics, HSA.APR.D.7: add, subtract, multiply and divide rational expressions
  • One pump fills a tank in 6 hours and a second pump fills it in 3 hours. How long do they take working together? Common Core State Standards for Mathematics, HSA.CED.A.1: create equations and inequalities in one variable and use them to solve problems
  • Simplify (1 + 1/x)/(1 − 1/x). Common Core State Standards for Mathematics, HSA.APR.D.6: rewrite simple rational expressions in different forms
  • Simplify (x² − 16)/(x² + 7x + 12). Common Core State Standards for Mathematics, HSA.APR.D.6: rewrite simple rational expressions in different forms
  • Solve (x + 2)/(x − 1) = 4. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Solve 1/x + 1/6 = 1/4. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Solve 3/x = 6/(x + 4). Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Solve x/(x − 5) = 5/(x − 5) + 3. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Subtract 5/(x − 2) − 3/(x + 2). Common Core State Standards for Mathematics, HSA.APR.D.7: add, subtract, multiply and divide rational expressions
  • What is the horizontal asymptote of f(x) = (6x − 1)/(2x + 3)? Common Core State Standards for Mathematics, HSF.IF.C.7.d: graph rational functions, identifying zeros and asymptotes
  • What is the vertical asymptote of f(x) = (2x + 1)/(x − 5)? Common Core State Standards for Mathematics, HSF.IF.C.7.d: graph rational functions, identifying zeros and asymptotes
  • y varies inversely with x, and y = 10 when x = 6. What is y when x = 15? Common Core State Standards for Mathematics, HSA.CED.A.1: create equations and inequalities in one variable and use them to solve problems

Radicals and rational exponents 16

  • At which point does the graph of y = √(x − 3) + 2 begin? Common Core State Standards for Mathematics, HSF.IF.C.7.b: graph square root, cube root and piecewise-defined functions
  • Evaluate 25^(−1/2). Common Core State Standards for Mathematics, HSN.RN.A.1: the meaning of rational exponents follows from the properties of integer exponents
  • Evaluate 27^(2/3). Common Core State Standards for Mathematics, HSN.RN.A.1: the meaning of rational exponents follows from the properties of integer exponents
  • Rationalize the denominator of 14/√7. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents
  • Simplify (8x⁶)^(1/3) for x ≥ 0. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents
  • Simplify √(48x²) for x ≥ 0. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents
  • Simplify √6 · √15. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents
  • Simplify √20 + √45. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents
  • Simplify √150. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents
  • Solve x^(3/2) = 27. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Solve √(2x − 1) = x − 2. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Solve √(x − 7) = 5. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • Solve ∛(x − 4) = 3. Common Core State Standards for Mathematics, HSA.REI.A.2: solve simple rational and radical equations in one variable, and see how extraneous solutions may arise
  • What is the domain of f(x) = √(x − 6)? Common Core State Standards for Mathematics, HSF.IF.B.5: relate the domain of a function to its graph
  • What is the inverse of f(x) = x³ + 2? Common Core State Standards for Mathematics, HSF.BF.B.4.a: solve f(x) = c for a simple function that has an inverse and write an expression for the inverse
  • Write ∛(x²) with a rational exponent, for x > 0. Common Core State Standards for Mathematics, HSN.RN.A.2: rewrite expressions involving radicals and rational exponents using the properties of exponents

Exponential and logarithmic functions 18

  • $1,000 is invested at 5% interest compounded once a year. What is it worth after 2 years? Common Core State Standards for Mathematics, HSF.LE.A.1.c: recognize a quantity that grows or decays by a constant percent rate per unit interval
  • A culture of 300 bacteria doubles every 4 hours. How many bacteria are there after 20 hours? Common Core State Standards for Mathematics, HSF.LE.A.2: construct linear and exponential functions, including arithmetic and geometric sequences
  • A sample of 80 grams has a half-life of 10 days. How much is left after 30 days? Common Core State Standards for Mathematics, HSF.LE.A.1.c: recognize a quantity that grows or decays by a constant percent rate per unit interval
  • Evaluate ln(e⁷). Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Evaluate log(5) + log(20). Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Evaluate log₂(128). Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Evaluate log₄(8). Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Evaluate log₅(1/25). Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • In y = 200(0.85)^t, by what percent does y fall each time t goes up by 1? Common Core State Standards for Mathematics, HSF.IF.C.8.b: use the properties of exponents to interpret expressions for exponential functions
  • Solve 2^x = 64. Common Core State Standards for Mathematics, HSF.LE.A.4: express as a logarithm the solution to an exponential equation and evaluate the logarithm
  • Solve 8^x = 32. Common Core State Standards for Mathematics, HSF.LE.A.4: express as a logarithm the solution to an exponential equation and evaluate the logarithm
  • Solve e^(2x) = 9. Which is the exact solution? Common Core State Standards for Mathematics, HSF.LE.A.4: express as a logarithm the solution to an exponential equation and evaluate the logarithm
  • Solve log₃(x) = 4. Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • What is the horizontal asymptote of y = 2^x − 3? Common Core State Standards for Mathematics, HSF.IF.C.7.e: graph exponential, logarithmic and trigonometric functions, showing intercepts, end behavior, period, midline and amplitude
  • What is the inverse of f(x) = 2^x? Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Which equation says the same thing as 3⁴ = 81? Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Which expression equals log₂(8x) for x > 0? Common Core State Standards for Mathematics, HSF.BF.B.5: the inverse relationship between exponents and logarithms
  • Which function shows exponential decay? Common Core State Standards for Mathematics, HSF.LE.A.1.c: recognize a quantity that grows or decays by a constant percent rate per unit interval

Sequences and series 14

  • A sequence has a₁ = 3 and aₙ = 2aₙ₋₁ − 1. What is a₄? Common Core State Standards for Mathematics, HSF.IF.A.3: sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
  • A sequence has a₁ = 4 and aₙ = aₙ₋₁ + 6. What is a₅? Common Core State Standards for Mathematics, HSF.IF.A.3: sequences are functions, sometimes defined recursively, whose domain is a subset of the integers
  • A theater has 18 seats in its first row and 3 more seats in each row after it. How many seats are in its 12 rows in all? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • In an arithmetic sequence, a₃ = 10 and a₇ = 26. What is the common difference? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • What is the 8th term of the geometric sequence 3, 6, 12, 24, ...? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • What is the 20th term of the arithmetic sequence 7, 11, 15, 19, ...? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • What is the common difference of the arithmetic sequence 23, 17, 11, 5, ...? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • What is the common ratio of the geometric sequence 81, 27, 9, 3, ...? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • What is the sum 2 + 6 + 18 + 54 + 162? Common Core State Standards for Mathematics, HSA.SSE.B.4: use the formula for the sum of a finite geometric series to solve problems
  • What is the sum of the first 6 terms of the geometric sequence 5, 10, 20, 40, ...? Common Core State Standards for Mathematics, HSA.SSE.B.4: use the formula for the sum of a finite geometric series to solve problems
  • What is the sum of the first 20 terms of the arithmetic sequence 2, 5, 8, 11, ...? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • Which explicit formula gives the geometric sequence 2, 10, 50, 250, ...? Common Core State Standards for Mathematics, HSF.LE.A.2: construct linear and exponential functions, including arithmetic and geometric sequences
  • Which explicit formula gives the sequence 5, 8, 11, 14, ...? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula
  • Which of these sequences is geometric? Common Core State Standards for Mathematics, HSF.BF.A.2: write arithmetic and geometric sequences both recursively and with an explicit formula

Trigonometric functions 10

  • Convert 3π/4 radians to degrees. Common Core State Standards for Mathematics, HSF.TF.A.1: radian measure of an angle as the length of the arc on the unit circle
  • Convert 60° to radians. Common Core State Standards for Mathematics, HSF.TF.A.1: radian measure of an angle as the length of the arc on the unit circle
  • How many radians are in one full turn around the unit circle? Common Core State Standards for Mathematics, HSF.TF.A.1: radian measure of an angle as the length of the arc on the unit circle
  • sin θ = 3/5 and θ is in Quadrant II. What is cos θ? Common Core State Standards for Mathematics, HSF.TF.C.8: use the Pythagorean identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ) or tan(θ)
  • The terminal side of an angle θ crosses the unit circle at (−3/5, 4/5). What is sin θ? Common Core State Standards for Mathematics, HSF.TF.A.2: the unit circle extends the trigonometric functions to all real numbers
  • What is cos(π)? Common Core State Standards for Mathematics, HSF.TF.A.2: the unit circle extends the trigonometric functions to all real numbers
  • What is sin(π/6)? Common Core State Standards for Mathematics, HSF.TF.A.3: use special triangles to determine the values of sine, cosine and tangent for π/3, π/4 and π/6
  • What is the amplitude of y = 4 sin(2x)? Common Core State Standards for Mathematics, HSF.TF.B.5: trigonometric functions with specified amplitude, frequency and midline
  • What is the midline of y = 3 sin(x) + 5? Common Core State Standards for Mathematics, HSF.TF.B.5: trigonometric functions with specified amplitude, frequency and midline
  • What is the period of y = cos(x/3)? Common Core State Standards for Mathematics, HSF.IF.C.7.e: graph exponential, logarithmic and trigonometric functions, showing intercepts, end behavior, period, midline and amplitude

How this list was compiled: Our own questions. The seven units are drawn from the high school Number and Quantity, Algebra and Functions standards of the Common Core State Standards for Mathematics (https://www.thecorestandards.org/Math/, read 2026-10-09). The standards name Algebra II in a traditional course sequence and say they do not mandate the sequence of high school courses (https://www.thecorestandards.org/Math/Content/note-on-courses-transitions/courses-transitions/, read 2026-10-09), so the grouping into units is this site's own. Each question names the standard it practices and links to that standard's page. Every answer is algebra or arithmetic on what the question prints, worked out a second time on 2026-10-09. Licence: Own work. Questions, answer choices, explanations and the graph were written and drawn for this site. Nothing is copied or reworded from a state end-of-course test, a school final, a placement test, a released form, a textbook or another practice test.

How it works

This free Algebra II practice test has 108 questions of our own, to answer on screen or print with a key. Every school and state writes its own Algebra 2 test, and this page copies none.

Read the full guide

A mixed test of 25

Exam mode draws 25 questions across the units and scores each unit, which makes it an Algebra 2 final exam practice test: the whole course in one sitting. The 25 is our number.

Where the units come from

Each question practices a high school standard of the Common Core State Standards for Mathematics, read October 9, 2026. The standards do not say which course holds which topic, so the grouping into units is ours.

  • Squaring both sides can add a false answer, so test every solution of a radical equation in the original.
  • A caret marks an exponent that cannot be typed raised, as in 27^(2/3) or 2^x, and log with no base shown is base 10.

Questions and answers

What is on an Algebra 2 test?

Your school's syllabus is the real list. The seven units here are quadratics with complex solutions, polynomial, rational and radical functions, exponentials and logarithms, sequences and series, and a first look at trigonometric functions. Matrices, conic sections and statistics are left out.

Is there an Algebra 2 practice test with answers PDF?

Yes. Print the question sheet opens 25 questions across the seven units, with the answer key if you tick it. Choose Save as PDF in the print window to keep the Algebra 2 practice test with answers as a file.

Is this an Algebra 2 EOC practice test?

Only as general practice. An end-of-course test is written by the state that gives it, to that state's own blueprint, and this page follows no state's blueprint. Your state education department's testing page says what its test covers.

Can I use this as an intermediate algebra practice test?

As general practice, yes. Each college sets what its intermediate algebra course covers, so compare the seven units with your syllabus and answer the ones it lists.

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