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Quadratics Checkpoint

A free College Algebra lesson from the “Quadratic Functions and Equations” unit, with a worked example and practice problems including step-by-step solutions.

This checkpoint mixes quadratic products, factoring, graph features, solving strategies, and applications.

What you'll learn

Why it matters: This checkpoint verifies that students can connect quadratic structure across products, factors, graphs, equations, and applications. That transfer matters because real quadratic problems rarely announce which representation to use.

Worked example

Problem. Factor x^2 - 3x - 28.

  1. Find numbers that multiply to -28.
  2. -7 and 4 add to -3.
  3. Write (x - 7)(x + 4).

Answer: (x - 7)(x + 4)

Practice problems

1. Expand (x + 5)(x - 2).

Choices: x^2 + 3x - 10 · x^2 - 7x - 10 · x^2 + 10 · 2x + 3

Show solution
  1. Products: First identify exactly what the question is asking: Expand (x + 5)(x - 2).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms are -2x and 5x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 + 3x - 10

2. Factor x^2 + x - 30.

Choices: (x + 6)(x - 5) · (x - 6)(x + 5) · (x + 30)(x - 1) · (x + 3)(x - 10)

Show solution
  1. Factoring: First identify exactly what the question is asking: Factor x^2 + x - 30.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 6 and -5 add to 1.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x + 6)(x - 5)

3. The vertex of y = (x + 1)^2 - 4 is...

Choices: (-1, -4) · (1, -4) · (-1, 4) · (1, 4)

Show solution
  1. Graphing: First identify exactly what the question is asking: The vertex of y = (x + 1)^2 - 4 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Vertex form gives (h, k).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-1, -4)

4. For h(t) = -t^2 + 8t, when does the maximum occur?

Show solution
  1. Applications: First identify exactly what the question is asking: For h(t) = -t^2 + 8t, when does the maximum occur?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Halfway between zeros 0 and 8.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

5. Solve x^2 - 11x + 30 = 0.

Choices: x = 5 or x = 6 · x = -5 or x = -6 · x = 3 or x = 10 · No real solution

Show solution
  1. Solving: First identify exactly what the question is asking: Solve x^2 - 11x + 30 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x - 5)(x - 6).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 5 or x = 6

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