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Quizzes > Mathematics

Free Algebra Quiz: Check Your Skills in Equations and Functions

Quick, free algebra test online with instant feedback and helpful hints.

Editorial: Review CompletedCreated By: Andrew GaulinUpdated Aug 27, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
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This free algebra quiz helps you practice equations, functions, and expressions, and shows where you might need more review. Target a skill with our solve for x quiz, build confidence with an algebra 1 quiz, or drill basics in simplifying expressions practice. Get instant feedback on each question.

What is the sum of 3x and 5x?
8x
15x
-2x
2x
When combining like terms, you add the coefficients of x. Here, 3 + 5 equals 8, so the result is 8x. Like terms must have the same variable raised to the same power to be combined. .
Solve the equation 2x + 5 = 15 for x.
x = -5
x = 10
x = 0
x = 5
Subtract 5 from both sides to get 2x = 10, then divide by 2 to find x = 5. Solving linear equations involves isolating the variable on one side. Always perform the same operation on both sides of the equation. .
Simplify the expression x^3 · x^2.
x
x^5
x^6
x^9
When multiplying expressions with the same base, you add the exponents: 3 + 2 = 5. Therefore, x^3 · x^2 = x^5. This is a basic exponent rule in algebra. .
Combine like terms: 4x^2 + 3x - 2x^2 + 7.
6x^2 + 3x + 7
2x^2 + 3x + 7
2x^2 + 3x - 7
4x^2 + x + 7
Combine the x^2 terms: 4x^2 - 2x^2 = 2x^2. The 3x term remains unchanged, and 7 is a constant. Thus the simplified form is 2x^2 + 3x + 7. .
Evaluate the function f(x) = 2x + 3 at x = 4.
11
9
7
8
Substitute x = 4 into the function: f(4) = 2(4) + 3 = 8 + 3 = 11. Function evaluation means replacing the variable with the given value. Introduction to function evaluation.
Expand the product (x + 2)(x + 3).
x^2 - 5x + 6
x^2 + 6x + 5
x^2 + 5x + 6
x^2 + 5x - 6
Use the distributive property (FOIL): x·x + x·3 + 2·x + 2·3 = x^2 + 3x + 2x + 6 = x^2 + 5x + 6. Expanding binomials is a key algebra skill. .
Solve for y: 3y - 9 = 0.
y = -3
y = 0
y = 6
y = 3
Add 9 to both sides to get 3y = 9, then divide by 3 to get y = 3. Isolating the variable involves reverse operations. .
Simplify the fraction (6x) / 2.
6x2
6/2 x
3x
x + 3
Divide the coefficient 6 by 2 to get 3, so the result is 3x. Simplifying numeric coefficients with variables follows the same rules as numbers. .
Combine like terms in 5a - 2a + 3.
5a + a + 3
3a - 3
3a + 3
7a + 3
Subtract the coefficients: 5a - 2a = 3a, then add the constant 3. The result is 3a + 3. Only like terms (same variable and exponent) combine. .
Solve for x: x/5 + 2 = 6.
10
-20
20
18
Subtract 2 from both sides: x/5 = 4, then multiply both sides by 5: x = 20. Isolating x step by step solves the linear equation. .
Solve the system: x + y = 5 and x - y = 1.
(5, 0)
(1, 4)
(2, 3)
(3, 2)
Add the equations to eliminate y: (x + y) + (x - y) = 5 + 1 gives 2x = 6, so x = 3. Substitute x back to find y = 2. .
Factor the quadratic expression x^2 - 5x + 6.
(x - 2)(x - 3)
(x + 2)(x + 3)
(x - 2)(x + 3)
(x - 1)(x - 6)
Find two numbers that multiply to 6 and add to -5: those are -2 and -3. Thus the factorization is (x - 2)(x - 3). .
What is the product of the roots of the equation x^2 - 4x - 5 = 0?
5
-1
4
-5
For ax^2 + bx + c = 0, the product of roots is c/a. Here a = 1 and c = -5, so the product is -5. This comes from Vieta's formulas. .
Simplify the radical ?50.
25?2
2?10
5?2
5?5
50 = 25·2, and ?25 = 5, so ?50 = 5?2. You look for perfect square factors to simplify radicals. .
Solve the inequality 2x - 3 < 7.
x > 2
x ? 5
x < 5
x > 5
Add 3 to both sides: 2x < 10, then divide by 2 to get x < 5. Inequalities reverse direction only when multiplied or divided by a negative. .
Convert y - 2 = 3(x + 1) to slope-intercept form.
y = 3x + 5
y = 3x - 5
y = -3x + 5
y = x + 5
Distribute 3: y - 2 = 3x + 3, then add 2: y = 3x + 5. Slope-intercept form is y = mx + b. .
Simplify the rational expression (x^2 - 9)/(x - 3).
x^2 + 3
x + 3
x - 3
(x - 3)/(x + 3)
Factor numerator: x^2 - 9 = (x - 3)(x + 3), then cancel (x - 3) to get x + 3. Domain excludes x = 3. .
Evaluate 2^3 · 2^-1.
1/4
8
4
2
Add exponents: 3 + (-1) = 2, so 2^3 · 2^-1 = 2^2 = 4. This follows the rule a^m · a^n = a^(m+n). .
Which set contains the solutions to |x - 4| = 3?
{1, 7}
{1, -7}
{4, 3}
{-1, 7}
Absolute value equation |x - 4| = 3 splits into x - 4 = 3 (x = 7) or x - 4 = -3 (x = 1). Thus the solutions are 1 and 7. .
What is the domain of f(x) = 1/(x - 2)?
x < 2
All real numbers
x > 2
All real numbers except x = 2
A function with a denominator cannot have that denominator equal zero. Setting x - 2 ? 0 gives x ? 2. All other real values are allowed. Domain of rational functions.
Solve the system: 2x + 3y = 6 and x - y = 4.
(4, -2)
(2, 2/5)
(6, -1.3)
(18/5, -2/5)
From x - y = 4, x = y + 4. Substitute into 2(y + 4) + 3y = 6 to get 2y + 8 + 3y = 6, so 5y = -2, y = -2/5, x = 18/5. .
What is the sum of the roots of x^2 + 6x + 5 = 0?
-5
5
6
-6
By Vieta's formula, the sum of the roots of ax^2 + bx + c = 0 is -b/a. Here b = 6 and a = 1, so the sum is -6. .
Evaluate log?(8).
4
3
2
1
log?(8) asks: 2 to what power equals 8? Since 2^3 = 8, log?(8) = 3. Logarithm definitions connect exponents and logs. .
Simplify (x^(1/2))^4.
x^2
x^(1/8)
x^4
x^3
Power of a power rule: (a^m)^n = a^(mn). Here m = 1/2 and n = 4, so x^(1/2 · 4) = x^2. .
Solve the inequality |2x - 1| ? 3.
x ? 2
x > -1
-1 ? x ? 2
1 ? x ? 2
Rewrite as -3 ? 2x - 1 ? 3. Add 1: -2 ? 2x ? 4, then divide by 2: -1 ? x ? 2. Absolute value inequalities split into two linear inequalities. .
Simplify (x^2 - 1)/(x^2 - 2x + 1).
x - 1
(x + 1)/(x - 1)
(x - 1)/(x + 1)
x + 1
Factor numerator as (x - 1)(x + 1) and denominator as (x - 1)^2, then cancel one (x - 1) to get (x + 1)/(x - 1). Domain excludes x = 1. .
What is the vertex of the parabola y = 2x^2 - 8x + 5?
(1, -1)
(4, -11)
(-2, 5)
(2, -3)
Vertex x-coordinate = -b/(2a) = 8/(4) = 2. Then y = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3, so (2, -3). .
Solve the quadratic inequality x^2 - 4x - 5 > 0.
x < -1 or x > 5
-1 < x < 5
x > 5 only
x > -1 and x < 5
Factor: (x - 5)(x + 1) > 0, so the product is positive when x < -1 or x > 5. Sketching a sign chart confirms intervals. .
Which value of x satisfies ?(x + 3) = x - 1?
No real solution
(3 - ?17)/2
Both (3 ± ?17)/2
(3 + ?17)/2
Square both sides: x + 3 = x^2 - 2x + 1 ? x^2 - 3x - 2 = 0. Solutions are (3 ± ?17)/2, but only (3 + ?17)/2 ? 1 satisfies the original. .
Which ordered pair (x, y) solves the nonlinear system x + y = 7 and xy = 10?
(2, 5)
(3, 4)
(10, -3)
(5, 2)
Substitute y = 7 - x into xy = 10: x(7 - x) = 10 ? x^2 - 7x + 10 = 0 ? (x - 5)(x - 2) = 0. So (5, 2) or (2, 5); as an ordered pair, (5, 2) is correct. .
What is the inverse function of f(x) = (3x - 2)/4?
(4x + 2)/3
(4x - 2)/3
(3x + 2)/4
(x + 2)/12
Swap x and y and solve: x = (3y - 2)/4 ? 4x = 3y - 2 ? 3y = 4x + 2 ? y = (4x + 2)/3. That is f?¹(x). Finding inverse functions.
Solve log??(x) + log??(x - 3) = 1 for x.
5
No valid solution
Both 5 and -2
-2
Combine logs: log??[x(x - 3)] = 1 ? x^2 - 3x = 10 ? x^2 - 3x - 10 = 0 ? (x - 5)(x + 2)=0. Only x=5 makes both logs defined. .
0
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Study Outcomes

  1. Understand fundamental algebraic concepts -

    Gain clarity on variables, constants, coefficients, and the structure of algebraic expressions and equations.

  2. Solve linear equations and inequalities -

    Develop step-by-step strategies to isolate variables, solve for unknowns, and verify your solutions in both equations and inequalities.

  3. Analyze functions and their properties -

    Explore function notation, identify domain and range, and interpret slopes and intercepts on their graphs.

  4. Apply algebraic techniques to real-world problems -

    Translate word problems into algebraic models and use appropriate methods to find practical solutions.

  5. Identify strengths and areas for improvement -

    Use your quiz results to pinpoint topics you've mastered and focus on skills that need further practice.

Cheat Sheet

  1. Mastering Linear Equations -

    Understanding how to isolate variables in one-step and two-step equations is crucial for success on any algebra quiz. For example, in 3x + 5 = 20, subtract 5 and divide by 3 to find x = 5, a method emphasized by resources like Khan Academy. Regular practice with basic algebra questions will boost your speed and accuracy for a free algebra quiz or algebra practice test.

  2. Applying the Quadratic Formula -

    The quadratic formula x = ( - b ± √(b² - 4ac)) / (2a) solves ax² + bx + c = 0 every time; a handy mnemonic is "minus b, plus or minus the square root." According to MIT OpenCourseWare, practicing a variety of quadratic problems helps you recognize when to factor versus using the formula. Mastery here ensures you breeze through related algebra challenge questions.

  3. Efficient Factoring Techniques -

    Factoring polynomials, such as using a² - b² = (a + b)(a - b) or the "ac method" for trinomials, underpins many algebra problems. Purplemath highlights the FOIL acronym (First, Outside, Inside, Last) to remember binomial multiplication in reverse. Sharpening this skill will make factoring pop up less intimidating on any algebra practice test.

  4. Understanding Functions & Graphs -

    Recognizing function notation f(x) and graphing lines in slope-intercept form y = mx + b helps you tackle function questions with confidence. The American Mathematical Society notes that identifying slope (m) and y-intercept (b) at a glance is a game-changer for quick graph sketches. Integrate these tips to ace the function sections of your free algebra quiz.

  5. Rules of Exponents & Radicals -

    Memorize key rules like aᵝ · a❿ = aᵝ❺❿ and (aᵝ)❿ = aᵝ❿ to simplify expressions swiftly. The University of Cambridge's math page suggests practicing negative and fractional exponents to demystify radicals (e.g., a² = √(a❴)). Consistent review of these laws makes exponent problems on an algebra challenge feel like second nature.

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