Use these Algebra 2 practice questions to review systems of equations, work-rate problems, rational equations, factoring, quadratic equations, completing the square, rational functions, linear functions, and logarithms. After answering each question, open the explanation to see the step-by-step solution.
Algebra 2 Topics Covered
- Solving systems of equations
- Working with rates and time
- Solving rational equations
- Factoring polynomials
- Using the quadratic formula
- Completing the square
- Finding vertical asymptotes
- Finding the domain of a function
- Solving logarithmic equations
Sample Questions
y = −3x + 4
x + 4y = −6
- A. x = −2, y = −1
- B. x = −2, y = 10
- C. x = 2, y = −2
- D. x = 3, y = −5
- E. x = 4, y = −8
Show Answer
Since the first equation already has y isolated, use substitution.
x + 4y = −6
Substitute −3x + 4 for y:
x + 4(−3x + 4) = −6
x − 12x + 16 = −6
−11x = −22
x = 2
Now substitute 2 for x in the first equation:
y = −3(2) + 4
y = −6 + 4
y = −2
The solution is x = 2, y = −2.
- A. 1 hour 12 minutes
- B. 1 hour 15 minutes
- C. 1 hour 20 minutes
- D. 1 hour 30 minutes
- E. 1 hour 35 minutes
Show Answer
John mows at a rate of
1
3
lawn per hour.
Julie mows at a rate of
1
2
lawn per hour.
Add their rates:
1
3
+
1
2
=
5
6
Together, they mow
5
6
of the lawn per hour.
To mow one whole lawn:
5
6
t = 1
t =
6
5
hours
6
5
hours is 1.2 hours, which is 1 hour 12 minutes.
5
x
−
3
x + 4
= 2
- A. −5
- B. −5 and 2
- C. 2
- D. 2 and 4
- E. 4
Show Answer
Multiply both sides by x(x + 4) to eliminate the denominators.
5(x + 4) − 3x = 2x(x + 4)
5x + 20 − 3x = 2x2 + 8x
2x + 20 = 2x2 + 8x
0 = 2x2 + 6x − 20
0 = x2 + 3x − 10
0 = (x + 5)(x − 2)
x = −5 or x = 2
Neither value makes a denominator equal zero, so both solutions are valid.
6x3 − 4x2 − 16x
- A. 0
- B. 2x(3x2 − 2x − 8)
- C. 2x(3x + 4)(x − 2)
- D. 4x(2x + 1)(x − 4)
- E. 2x(2x2 + 7x − 4)
Show Answer
First, factor out the greatest common factor.
6x3 − 4x2 − 16x = 2x(3x2 − 2x − 8)
Now factor the trinomial:
3x2 − 2x − 8 = (3x + 4)(x − 2)
So the complete factorization is:
2x(3x + 4)(x − 2)
5x2 + 6x = 3
- A.
−6 ± √22
5
- B.
−3 ± √22
5
- C.
−3 ± 2√6
5
- D.
3 ± 2√6
5
- E.
6 ± 2√6
5
Show Answer
−3 ± 2√6
5
First, move all terms to one side:
5x2 + 6x − 3 = 0
Use the quadratic formula:
x =
−b ± √(b2 − 4ac)
2a
Here, a = 5, b = 6, and c = −3.
x =
−6 ± √(62 − 4(5)(−3))
2(5)
x =
−6 ± √96
10
Since √96 = 4√6, simplify:
x =
−6 ± 4√6
10
=
−3 ± 2√6
5
- A. −36
- B. −12x
- C. −6
- D. 12x
- E. 36
Show Answer
To complete the square, take half of the coefficient of x, then square the result.
The coefficient of x is −12.
Half of −12 is −6.
(−6)2 = 36
So 36 should be added to both sides.
x2 − 12x + 36 = 5 + 36
The left side becomes a perfect-square trinomial:
(x − 6)2 = 41
y =
x2 − 36
x2 − 8x + 15
- A. x = −5 and x = −3
- B. x = −5, x = −3, and x = 6
- C. x = 3 and x = 5
- D. x = 3 and x = 6
- E. x = 6
Show Answer
For a rational function, vertical asymptotes occur where the denominator equals zero, as long as the factor does not cancel with the numerator.
Set the denominator equal to zero:
x2 − 8x + 15 = 0
Factor the denominator:
(x − 3)(x − 5) = 0
x = 3 or x = 5
The numerator factors as (x − 6)(x + 6), so no factors cancel with the denominator.
Therefore, the vertical asymptotes are x = 3 and x = 5.
- A. 1 hour 12 minutes
- B. 2 hours
- C. 5 hours
- D. 6 hours
- E. 6 hours 40 minutes
Show Answer
Let t represent the number of hours the second car travels.
The first car has been driving for 3 hours longer, so its time is t + 3.
Distance = rate × time
First car’s distance: 40(t + 3)
Second car’s distance: 60t
The second car catches the first when the distances are equal:
40(t + 3) = 60t
40t + 120 = 60t
120 = 20t
t = 6
The second car catches the first 6 hours after it leaves.
- A.
x ≥ −
1
2
- B.
x < −
1
2
- C. x > 0
- D.
x ≥
1
2
- E. All real numbers
Show Answer
The domain is the set of all possible input values.
The function f(x) = 2x − 4 is linear.
It does not contain a square root, logarithm, or denominator that could restrict the value of x.
Therefore, the domain is all real numbers.
log2
7x + 3
x
= 3
- A. −3
- B.
−
1
2
- C. 2
- D. 3
- E. 5
Show Answer
A logarithm can be rewritten in exponential form.
Since log2 of the expression equals 3, the expression must equal 23.
7x + 3
x
= 8
7x + 3 = 8x
3 = x
So x = 3.
How to Use These Algebra 2 Practice Questions
Start by answering each question before opening the explanation. Then compare your work to the step-by-step solution. If you miss a question, review the related Algebra 2 skill before moving on.
For extra review, focus on the topics you miss most often. Algebra 2 questions often depend on recognizing the correct setup, such as substitution, factoring, the quadratic formula, completing the square, or rewriting logarithmic equations.