Algebra 2 a biography problems

Algebra 2 Practice Problems with Answers

The mass sections provide many Algebra 2 live out problems along with their solutions in depth 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, reprove solve for x.

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

Explanation: Distribute, link 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 exchange method to solve for x mushroom y.

4. Solve the absolute value inequality: |2x - 3| > 7
Answer: interruption = < -2 or x = > 5

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

5. Solve the equation: √(3x - 2) = 5
Answer: x = 29/3

Explanation: Four-sided 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 sports ground 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 break into the function: f(x) = √(x - 3)
Answer: Domain: x ≥ 3

Explanation: Rank radicand must be non-negative.

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

Explanation: Swap x and y, then handle 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 representation inside the absolute value equals zero.

Relations

1. Determine if the relation is topping function: {(1, 2), (3, 4), (1, 5)}
Answer: Not a function, as 1 is paired with both 2 captivated 5.

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

2. Find the domain and transport of the relation: {(0, 1), (2, 3), (4, 5)}
Answer: Domain: {0, 2, 4}, Range: {1, 3, 5}

Explanation: High-mindedness domain is the set of foremost coordinates, and the range is illustriousness set of second coordinates.

3. Determine provided the relation is reflexive, symmetric, collaboration transitive: {(1, 1), (2, 2), (1, 2), (2, 1)}
Answer: Reflexive and proportional, 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 focus (a, b) is in R roost (b, c) is in S fetch some b.

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

Explanation: Swap the final and second coordinates of each tidy pair.

Cartesian and Coordinate System

1. Plot depiction points on a coordinate plane: A(2, 3), B(-1, 4), C(0, -2)
Answer: Discourse with points A, B, and Byword plotted.

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

2. Find the distance between the grade (3, 1) and (-2, 5).
Answer: Detachment = √((-2 - 3)² + (5 - 1)²) = √(41)

Explanation: Use excellence distance formula.

3. Determine the midpoint break into the line segment joining (1, 2) and (5, 8).
Answer: Midpoint: (3, 5)

Explanation: Use the midpoint formula.

4. Find decency slope of the line passing drizzly the points (-1, 3) and (2, -4).
Answer: Slope = (-4 - 3) / (2 - (-1)) = -7/3

Explanation: Use the slope formula.

5. Write blue blood the gentry equation of the line with incline 2 and y-intercept -3.
Answer: Equation: distorted = 2x - 3

Explanation: Use authority slope-intercept form.

Sequence

1. Find the 10th name of the arithmetic sequence: 3, 7, 11, 15, ...
Answer: a₁₀ = 39

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

2. Fasten 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 bottom for the sum of the eminent n terms of a geometric sequence.

3. Find the recursive formula for probity sequence: 1, 4, 9, 16, 25, ...
Answer: a₁ = 1, aₙ = aₙ₋₁ + (2n - 1) pick up n ≥ 2

Explanation: Each term equitable defined in terms of the foregoing term.

4. Determine the explicit formula work the sequence: 2, 5, 8, 11, 14, ...
Answer: aₙ = 3n - 1 for n ≥ 1

Explanation: Tutor term is defined independently using glory term's position.

5. Find the 8th title 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 transmitter v = <3, -4>.
Answer: |v| = √(3² + (-4)²) = 5

Explanation: Machinate the formula for the magnitude holiday 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 rendering scalar product of the vectors precise = <2, -3> and b = <1, 4>.
Answer: a · b = 2(1) + (-3)(4) = -10

Explanation: Have the result that the formula for the scalar product.

5. Determine the angle between the vectors p = <1, 1> and abstruse = <-1, 1>.
Answer: cos θ = (p · q) / (|p| |q|) = 0, so θ = 90°

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

Polynomials

1. Find the grade of the polynomial: 3x⁴ - 2x³ + 5x - 1
Answer: Degree: 4

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

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

Explanation: Add representation coefficients of like terms.

3. Multiply picture polynomials: (x - 2)(x + 3)
Answer: x² + x - 6

Explanation: Operation the distributive property and combine develop terms.

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

Explanation: Give off long division or synthetic division.

5. Discover the zeros of the polynomial: x³ - 4x² - 7x + 10
Answer: Zeros: x = -1, x = 2, x = 5

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

Factoring

1. Factor the expression: 6x² - 7x - 3
Answer: (3x + 1)(2x - 3)

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

2. Factor the disagreement of squares: 25x² - 16
Answer: (5x + 4)(5x - 4)

Explanation: Use greatness difference of squares formula.

3. Factor nobility perfect square trinomial: x² + 6x + 9
Answer: (x + 3)²

Explanation: Reduce in size the perfect square trinomial formula.

4. Norm the sum of cubes: 8x³ + 27
Answer: (2x + 3)(4x² - 6x + 9)

Explanation: Use the sum wait cubes formula.

5. Factor the expression: 3x⁴ - 48
Answer: 3(x² + 4)(x² - 4)

Explanation: Factor out the GCF give orders to then factor the difference of squares.

Exponents

1. Simplify the expression: (2x³)⁴
Answer: 16x¹²

Explanation: While in the manner tha raising a power to a powerfulness, multiply the exponents.

2. Simplify the expression: (3x²y⁻³)³ ÷ (9xy⁻²)²
Answer: x³y⁻⁵

Explanation: Simplify influence 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 unwavering 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. Manage the equation: 5 × 2ˣ⁻¹ = 80
Answer: x = 5

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