## Algebra 2 Practice Problems with Answers

The following sections provide many Algebra 2 practice problems along with their solutions to help you master the key concepts of the course.

### Linear Equations and Inequalities

1. Solve for x: 3(2x - 5) + 4 = 19

**Answer**: x = 5

Explanation: Distribute, combine like terms, and solve for x.

2. Solve the inequality: 2(3x + 1) ≤ 5x - 7

**Answer**: x = -1

Explanation: Distribute, combine like terms, and solve the inequality.

3. Solve the system of equations: 2x + 3y = 11 x - y = 1

**Answer**: x = 3, y = 2

Explanation: Use the substitution method to solve for x and y.

4. Solve the absolute value inequality: |2x - 3| > 7

**Answer**: x = < -2 or x = > 5

Explanation: Isolate the absolute value term and solve two separate inequalities.

5. Solve the equation: √(3x - 2) = 5

**Answer**: x = 29/3

Explanation: Square both sides and solve for x.

### Functions

1. Given f(x) = 2x² - 5x + 3, find f(-2).

**Answer**: f(-2) = 19

Explanation: Substitute -2 for x and simplify.

2. If f(x) = 3x - 1 and g(x) = x² + 2, find (f ∘ g)(x).

**Answer**: (f ∘ g)(x) = 3(x² + 2) - 1 = 3x² + 5

Explanation: Substitute g(x) for x in f(x) and simplify.

3. Determine the domain of the function: f(x) = √(x - 3)

**Answer**: Domain: x ≥ 3

Explanation: The radicand must be non-negative.

4. Find the inverse of the function: f(x) = (2x + 1) / 3

**Answer**: f⁻¹(x) = (3x - 1) / 2

Explanation: Swap x and y, then solve for y.

5. Graph the function: f(x) = |x - 2| + 1

**Answer**: V-shaped graph with vertex at (2, 1)

Explanation: The graph is a V-shape with the vertex where the expression inside the absolute value equals zero.

### Relations

1. Determine if the relation is a function: {(1, 2), (3, 4), (1, 5)}

**Answer**: Not a function, as 1 is paired with both 2 and 5.

Explanation: In a function, each x-value is paired with at most one y-value.

2. Find the domain and range of the relation: {(0, 1), (2, 3), (4, 5)}

**Answer**: Domain: {0, 2, 4}, Range: {1, 3, 5}

Explanation: The domain is the set of first coordinates, and the range is the set of second coordinates.

3. Determine if the relation is reflexive, symmetric, or transitive: {(1, 1), (2, 2), (1, 2), (2, 1)}

**Answer**: Reflexive and symmetric, but not transitive.

Explanation: Check the definitions of reflexive, symmetric, and transitive relations.

4. Compose the relations: R = {(1, 2), (2, 3)} and S = {(2, 4), (3, 5)}

**Answer**: R ∘ S = {(1, 4), (2, 5)}

Explanation: Find pairs (a, c) such that (a, b) is in R and (b, c) is in S for some b.

5. Find the inverse of the relation: {(1, 3), (2, 4), (5, 6)}

**Answer**: Inverse: {(3, 1), (4, 2), (6, 5)}

Explanation: Swap the first and second coordinates of each ordered pair.

### Cartesian and Coordinate System

1. Plot the points on a coordinate plane: A(2, 3), B(-1, 4), C(0, -2)

**Answer**: Graph with points A, B, and C plotted.

Explanation: Find the x-coordinate on the horizontal axis and the y-coordinate on the vertical axis for each point.

2. Find the distance between the points (3, 1) and (-2, 5).

**Answer**: Distance = √((-2 - 3)² + (5 - 1)²) = √(41)

Explanation: Use the distance formula.

3. Determine the midpoint of the line segment joining (1, 2) and (5, 8).

**Answer**: Midpoint: (3, 5)

Explanation: Use the midpoint formula.

4. Find the slope of the line passing through the points (-1, 3) and (2, -4).

**Answer**: Slope = (-4 - 3) / (2 - (-1)) = -7/3

Explanation: Use the slope formula.

5. Write the equation of the line with slope 2 and y-intercept -3.

**Answer**: Equation: y = 2x - 3

Explanation: Use the slope-intercept form.

### Sequence

1. Find the 10th term of the arithmetic sequence: 3, 7, 11, 15, ...

**Answer**: a₁₀ = 39

Explanation: Use the formula for the nth term of an arithmetic sequence.

2. Determine the sum of the first 20 terms of the geometric sequence: 2, 6, 18, 54, ...

**Answer**: S₂₀ = 2(3²⁰ - 1) / (3 - 1) = 3,486,784,400

Explanation: Use the formula for the sum of the first n terms of a geometric sequence.

3. Find the recursive formula for the sequence: 1, 4, 9, 16, 25, ...

**Answer**: a₁ = 1, aₙ = aₙ₋₁ + (2n - 1) for n ≥ 2

Explanation: Each term is defined in terms of the preceding term.

4. Determine the explicit formula for the sequence: 2, 5, 8, 11, 14, ...

**Answer**: aₙ = 3n - 1 for n ≥ 1

Explanation: Each term is defined independently using the term's position.

5. Find the 8th term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ...

**Answer**: F₈ = 21

Explanation: Use the recursive formula to calculate each term successively.

### Vector

1. Find the magnitude of the vector v = <3, -4>.

**Answer**: |v| = √(3² + (-4)²) = 5

Explanation: Use the formula for the magnitude of a vector.

2. Add the vectors u = <2, 1> and v = <-1, 3>.

**Answer**: u + v = <1, 4>

Explanation: Add the corresponding components.

3. Subtract the vector v = <4, -2> from u = <1, 5>.

**Answer**: u - v = <-3, 7>

Explanation: Subtract the corresponding components.

4. Find the scalar product of the vectors a = <2, -3> and b = <1, 4>.

**Answer**: a · b = 2(1) + (-3)(4) = -10

Explanation: Use the formula for the scalar product.

5. Determine the angle between the vectors p = <1, 1> and q = <-1, 1>.

**Answer**: cos θ = (p · q) / (|p| |q|) = 0, so θ = 90°

Explanation: Use the formula for the angle between two vectors.

### Polynomials

1. Find the degree of the polynomial: 3x⁴ - 2x³ + 5x - 1

**Answer**: Degree: 4

Explanation: The degree is the highest power of the variable.

2. Add the polynomials: (2x² - 3x + 1) + (x² + 4x - 2)

**Answer**: 3x² + x - 1

Explanation: Add the coefficients of like terms.

3. Multiply the polynomials: (x - 2)(x + 3)

**Answer**: x² + x - 6

Explanation: Use the distributive property and combine like terms.

4. Divide the polynomials: (2x³ - 5x² + 3x - 1) ÷ (x - 1)

**Answer**: Quotient: 2x² - 3x + 3, Remainder: 2

Explanation: Use long division or synthetic division.

5. Find the zeros of the polynomial: x³ - 4x² - 7x + 10

**Answer**: Zeros: x = -1, x = 2, x = 5

Explanation: Factor the polynomial and set each factor equal to zero.

### Factoring

1. Factor the expression: 6x² - 7x - 3

**Answer**: (3x + 1)(2x - 3)

Explanation: Find two numbers whose product is ac and whose sum is b.

2. Factor the difference of squares: 25x² - 16

**Answer**: (5x + 4)(5x - 4)

Explanation: Use the difference of squares formula.

3. Factor the perfect square trinomial: x² + 6x + 9

**Answer**: (x + 3)²

Explanation: Use the perfect square trinomial formula.

4. Factor the sum of cubes: 8x³ + 27

**Answer**: (2x + 3)(4x² - 6x + 9)

Explanation: Use the sum of cubes formula.

5. Factor the expression: 3x⁴ - 48

**Answer**: 3(x² + 4)(x² - 4)

Explanation: Factor out the GCF and then factor the difference of squares.

### Exponents

1. Simplify the expression: (2x³)⁴

**Answer**: 16x¹²

Explanation: When raising a power to a power, multiply the exponents.

2. Simplify the expression: (3x²y⁻³)³ ÷ (9xy⁻²)²

**Answer**: x³y⁻⁵

Explanation: Simplify the numerator and denominator separately, then divide.

3. Solve the equation: 4ˣ⁺¹ = 64

**Answer**: x = 2

Explanation: Set the exponents equal to each other and solve for x.

4. Simplify the expression: (27a⁶b⁻⁹)⅓ ÷ (9a²b⁻³)½

**Answer**: b⁻¹

Explanation: Simplify the numerator and denominator separately, then divide.

5. Solve the equation: 5 × 2ˣ⁻¹ = 80

**Answer**: x = 5

Explanation: Isolate the exponential term, then apply the logarithm to both sides.