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Quizzes > Mathematics > Algebra

Put Your Skills to the Test with the Algebra 1 Final Exam

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Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style algebra exam with equations functions and variable symbols on golden yellow background

Use this Algebra 1 final exam quiz to practice equations, functions, inequalities, and word problems so you can spot gaps before test day. Work at your pace with instant feedback and step-by-step help, then try the full practice exam or keep going with this algebra question set.

Simplify the expression 3x + 5x.
2x
15x
3+5x
8x
Combining like terms involves adding coefficients of the same variable. Here, 3x + 5x equals (3+5)x which is 8x. Like terms must have identical variable parts to be combined or added. For more on combining like terms, see .
Solve for x: x + 7 = 12.
5
-19
19
-5
To isolate x, subtract 7 from both sides of the equation: x = 12 - 7, giving x = 5. This one-step equation demonstrates the inverse operation of addition. Checking the solution in the original equation verifies correctness. For more practice, see .
Evaluate f(2) if f(x) = 3x + 4.
12
6
8
10
To evaluate f(2), substitute x = 2 into the expression: f(2) = 3(2) + 4 = 6 + 4 = 10. Function evaluation means replacing the input variable with the given value. Always perform multiplication before addition to maintain order of operations. See more at .
Simplify 2^3 × 2^4.
128
16
32
64
When multiplying powers with the same base, you add the exponents: 2^3 × 2^4 = 2^(3+4) = 2^7 = 128. This is a core exponent rule for simplifying expressions. Always ensure the bases are identical before adding exponents. For more, see .
What is the slope of the line passing through (1, 2) and (3, 6)?
-2
1/2
4
2
Slope is rise over run: (6 - 2) / (3 - 1) = 4/2 = 2. It measures the steepness of a line between two points. Always subtract y-values first, then x-values to get the correct ratio. For details, see .
Solve 2x - 5 = 9 for x.
-2
7
-7
14
Add 5 to both sides to get 2x = 14, then divide by 2 to find x = 7. This two-step linear equation uses inverse operations to isolate x. Always perform addition or subtraction before division when following order of operations in solving. More examples at .
What is the y-intercept of the line y = 4x - 3?
3
4
0
-3
In slope-intercept form y = mx + b, the constant term b is the y-intercept. Here, the line crosses the y-axis at (0, -3). This value shows where x is zero. Learn more at .
Combine like terms: 6a - 2a + 3a.
6a
7a
11a
a
Like terms share the same variable: 6a - 2a = 4a, then 4a + 3a = 7a. Combining like terms simplifies polynomial expressions. Ensure you only add or subtract terms with identical variables and exponents. See for more.
Use the distributive property to expand 3(x + 5).
3x + 5
15x + 3
x + 15
3x + 15
The distributive property multiplies each term inside the parentheses by the factor outside: 3×x + 3×5 = 3x + 15. This method removes parentheses and simplifies expressions. Check each multiplication carefully. More at .
Solve x/5 = 7.
35
2.8
-35
12
Multiply both sides by 5 to isolate x: x = 7 × 5 = 35. This operation undoes division by 5. Always apply the inverse operation to solve for the variable. For more examples, see .
Simplify the expression -(-x + 4).
x - 4
-x - 4
-x + 4
x + 4
Distribute the negative sign: -1×(-x) = x, and -1×4 = -4, giving x - 4. Removing parentheses with the correct sign is key. Verify by plugging in a value for x. See for more on distribution with negatives.
Solve the system: 2x + y = 8 and x - y = 2.
(10/3, 4/3)
(3, 2/3)
(2, 4)
(4, 2)
From x - y = 2, x = y + 2. Substitute into 2(y + 2) + y = 8 to get 3y + 4 = 8, so y = 4/3 and x = 10/3. Checking both equations confirms the solution. For substitution tutorials, see .
Multiply (x + 3)(x - 5).
x^2 - 15x + 15
x^2 + 15x - 15
x^2 + 2x - 15
x^2 - 2x - 15
Use FOIL: x·x + x·(-5) + 3·x + 3·(-5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15. Always multiply in the order First, Outer, Inner, Last. To practice, see .
Factor the trinomial x^2 - 6x + 9.
(x + 3)^2
(x - 3)^2
(x - 9)(x + 1)
(x - 3)(x + 3)
x^2 - 6x + 9 is a perfect square trinomial because (-3)^2 = 9 and 2·(-3) = -6. It factors to (x - 3)(x - 3), or (x - 3)^2. Recognizing patterns speeds factoring. More at .
Simplify the rational expression (3x^2)/(9x).
3x
x/3
x^2/9
1/3
Divide numerator and denominator by 3x: (3x^2)/(9x) = x/3. Cancel common factors to simplify rational expressions. Restrict x ? 0 to avoid division by zero. For more, see .
Solve the inequality 3x - 4 < 8.
x < 4
x ? 4
x ? 4
x > 4
Add 4 to both sides: 3x < 12, then divide by 3: x < 4. Inequality direction stays the same when dividing by a positive number. Graphing this shows all values less than 4. Learn more at .
Solve |x - 2| = 5.
x = 7 or x = -3
x = 3 or x = -1
x = 2 or x = -2
x = 5 or x = -5
Absolute value equations split into two cases: x - 2 = 5 ? x = 7, or x - 2 = -5 ? x = -3. Check both in the original equation. This method handles distance from zero definitions. See .
Find g(f(1)) if f(x) = 2x + 1 and g(x) = x^2.
4
9
5
7
First evaluate f(1): 2(1) + 1 = 3. Then g(3) = 3^2 = 9. Function composition means input the result of one function into the next. For more, see .
What is the domain of f(x) = ?(x - 2)?
x ? 2
All real x
x ? 2
x > 2
The expression under the square root must be nonnegative: x - 2 ? 0 ? x ? 2. Domain restrictions come from even roots. Values below 2 would make the radicand negative. See .
Simplify the expression x^0 + 5.
x + 5
5
1
6
By definition, any nonzero base to the zero power is 1: x^0 = 1 (as long as x ? 0). Therefore, x^0 + 5 = 1 + 5 = 6. Understanding exponent properties helps simplify expressions. More at .
Solve 4(x - 1) = 2x + 6.
-5
5
1
4
Expand left side: 4x - 4 = 2x + 6. Subtract 2x: 2x - 4 = 6; add 4: 2x = 10; divide by 2: x = 5. Always apply operations in inverse order to isolate the variable. See .
Convert y - 2 = 3(x + 1) to slope-intercept form and identify slope and y-intercept.
Slope = -3, y-intercept = 5
Slope = 5, y-intercept = 3
Slope = 3, y-intercept = -5
Slope = 3, y-intercept = 5
Distribute: y - 2 = 3x + 3 ? y = 3x + 5, so slope m = 3 and y-intercept b = 5. Rewrite into y = mx + b to read slope and intercept. Always perform distribution before isolating y. For more, see .
Solve x^2 + 6x + 5 = 0 by completing the square.
x = -1 or x = -5
x = 2 or x = 3
x = -2 or x = -4
x = 1 or x = 5
x^2 + 6x + 5 = 0 ? x^2 + 6x + 9 = 4 ? (x + 3)^2 = 4 ? x + 3 = ±2 ? x = -1 or -5. Completing the square involves adding (b/2)^2 to both sides. Check both solutions in the original equation. For steps, see .
Use the quadratic formula to solve 2x^2 - 8x + 6 = 0.
x = 1 or x = 3
x = -1 or x = -3
x = 1 or x = 2
x = 2 or x = 3
For ax^2+bx+c=0, x = [8 ± ?(64 - 48)]/(4) = (8 ± 4)/4 ? x = 3 or 1. Quadratic formula handles any quadratic equation. Always simplify under the radical before proceeding. More at .
Simplify (x^2 - 9)/(x - 3).
x - 3
x^2 + 3
x + 3
1
Factor numerator: (x - 3)(x + 3)/(x - 3) = x + 3, x ? 3. Cancelling common factors simplifies the expression. Watch for excluded values where the original denominator is zero. See .
Simplify 2x^3 × 3x^2.
6x^5
6x^6
5x^6
5x^5
Multiply coefficients: 2×3 = 6, and add exponents: 3 + 2 = 5 ? 6x^5. This is the product rule for exponents. Always handle numeric and variable parts separately. For more, see .
Factor the sum of cubes x^3 + 8.
(x + 2)(x^2 - 2x + 4)
(x - 2)(x^2 + 2x + 4)
(x + 2)^3
(x + 2)(x^2 + 2x + 4)
Sum of cubes a^3 + b^3 = (a + b)(a^2 - ab + b^2). Here a = x, b = 2, so (x + 2)(x^2 - 2x + 4). Recognizing special products aids factoring. More at .
Simplify (4x^-2)/(2x^-5).
8x^2
x^3/2
2x^3
2/x^3
Divide coefficients: 4/2 = 2, and subtract exponents: -2 - (-5) = 3 ? 2x^3. The quotient rule for exponents is used. Keep track of negative signs carefully. See .
Solve ?(x + 4) = 3.
9
-9
5
-5
Square both sides: x + 4 = 9 ? x = 5. Check that x + 4 ? 0; 5 + 4 = 9 is valid. Radical equations require checking for extraneous roots. More at .
Multiply (x/(x+2)) × ((x+2)/(x-1)).
x/(x-1)
(x+2)/(x-1)
x^2/(x^2+1)
(x-1)/x
Cancel the common factor (x + 2) and multiply numerators/denominators: x/(x-1). Excluded values: x ? -2, 1. Simplifying before multiplying prevents extra work. For more, see .
Factor 9x^2 - 16.
(3x - 4)(3x + 4)
(9x + 16)
9x^2 + 16
(3x - 4)^2
This is a difference of squares: (3x)^2 - 4^2 = (3x - 4)(3x + 4). Recognizing a^2 - b^2 = (a - b)(a + b) is key. Be careful with signs. See .
Simplify (x^(1/2))^4.
x^2
x^4
2x
x
Raise a power to a power by multiplying exponents: (x^(1/2))^4 = x^(1/2 × 4) = x^2. This is a basic exponent rule. Ensure to multiply fractions correctly. More at .
Solve the inequality x^2 - 4x + 3 ? 0.
x < 1 or x > 3
1 ? x ? 3
x ? 1
x ? 3
Factor: (x - 1)(x - 3) ? 0. The parabola opens up, so values between the roots inclusive satisfy the inequality: 1 ? x ? 3. Test points to confirm. For more, see .
Solve the absolute value inequality |2x - 3| > 5.
x ? -1 or x ? 4
x < -1 or x > 4
x > -1 and x < 4
-1 < x < 4
Split into two cases: 2x - 3 > 5 ? x > 4, and 2x - 3 < -5 ? x < -1. The solution set is x < -1 or x > 4. Always reverse the inequality when multiplying/dividing by negative in absolute value cases. See .
Solve for x: (x + 1)/(x - 2) = 3.
5/2
5/3
7/2
-7/2
Multiply both sides by (x - 2): x + 1 = 3x - 6 ? -2x = -7 ? x = 7/2. Exclude x = 2 from the domain because it makes the denominator zero. Always check for domain restrictions in rational equations. See .
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Study Outcomes

  1. Solve Linear Equations -

    Apply step-by-step techniques to solve one-variable and multi-step equations, ensuring accuracy on Algebra 1 final exam questions.

  2. Simplify Algebraic Expressions -

    Use properties of operations to combine like terms, distribute, and simplify expressions for efficient problem solving.

  3. Factor Polynomials -

    Identify and apply factoring methods, including greatest common factors and special products, to rewrite polynomials in factored form.

  4. Graph and Interpret Functions -

    Plot linear and nonlinear functions on a coordinate plane and interpret key features such as slope, intercepts, and curvature.

  5. Analyze Systems of Equations -

    Solve systems using substitution and elimination methods to find intersection points and verify solutions.

  6. Apply Quadratic Concepts -

    Use factoring, completing the square, and the quadratic formula to solve quadratic equations and understand their graphs.

Cheat Sheet

  1. Slope-Intercept Form Mastery -

    Review the slope-intercept formula y = mx + b, where m represents the rate of change and b is the starting value. For example, y = 2x + 3 describes a line rising two units every step right and crossing the y-axis at 3. This foundational concept is emphasized by the Common Core State Standards for Understanding Linear Functions.

  2. Solving Systems of Equations -

    Practice substitution and elimination methods to solve pairs of linear equations, like solving 2x + y = 5 and x - y = 1 simultaneously. Substitution isolates one variable first, while elimination adds equations to cancel a variable - choose whichever feels more intuitive. These techniques are widely taught in university algebra courses such as those at MIT OpenCourseWare.

  3. Factoring and the Quadratic Formula -

    Master factoring trinomials (ax² + bx + c) into (dx + e)(fx + g) and apply the zero-product property to find roots. When factoring is tricky, use the quadratic formula x = [ - b ± √(b² - 4ac)]/(2a) for a foolproof solution. This approach is validated by research in educational journals like the Journal of Mathematical Behavior for boosting problem-solving fluency.

  4. Exponents and Scientific Notation -

    Memorize exponent rules - such as a^m × a^n = a^(m+n) and (a^m)^n = a^(mn) - and apply them to simplify expressions quickly. Practice converting large or small numbers into scientific notation (e.g., 6.02×10^23) to handle data more efficiently, a skill endorsed by the National Institute of Standards and Technology. Use the mnemonic "Multiply bases? Add exponents!" to recall the laws under pressure.

  5. Functions, Domain, and Range -

    Understand that a function f(x) uniquely assigns each input x to one output y, and sketching its graph requires checking the vertical line test. Identify the domain (allowable x-values) and range (resulting y-values) by analyzing where the function is defined, for example f(x)=√(x - 2) has domain x ≥ 2. The National Council of Teachers of Mathematics highlights these concepts as essential for fluency in Algebra 1 final exam problems.

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