CMClearMathAcademy

Quadratic Deepening Checkpoint

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

This checkpoint reviews multiple quadratic strategies. The goal is to choose the method that fits the form and meaning of the problem.

What you'll learn

Why it matters: This checkpoint pulls together factoring, the quadratic formula, vertex form, completing the square, and the discriminant. Quadratics show up in nearly every applied scenario from physics to finance, which is why the deepening checkpoint earns its own slot.

Worked example

Problem. Find the vertex of y = -3(x + 2)^2 + 5.

  1. Use vertex form y = a(x - h)^2 + k.
  2. x + 2 means h = -2.
  3. The vertex is (-2, 5).

Answer: (-2, 5)

Practice problems

1. Factor x^2 - 3x - 18.

Choices: (x - 6)(x + 3) · (x + 6)(x - 3) · (x - 9)(x + 2) · (x - 18)(x + 1)

Show solution
  1. Factoring: First identify exactly what the question is asking: Factor x^2 - 3x - 18.
  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 3 multiply to -18 and add to -3.
  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 + 3)

2. Solve x^2 - 10x + 21 = 0.

Choices: 3 and 7 · -3 and -7 · 1 and 21 · 5 and 5

Show solution
  1. Solving: First identify exactly what the question is asking: Solve x^2 - 10x + 21 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x - 3)(x - 7).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3 and 7

3. The vertex of y = 2(x - 5)^2 - 6 is...

Choices: (5, -6) · (-5, -6) · (5, 6) · (-5, 6)

Show solution
  1. Vertex Form: First identify exactly what the question is asking: The vertex of y = 2(x - 5)^2 - 6 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: (5, -6)

4. What number completes the square for x^2 + 16x?

Show solution
  1. Completing Square: First identify exactly what the question is asking: What number completes the square for x^2 + 16x?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Half of 16 is 8.
  4. 8^2 = 64.
  5. Check the result by substituting or estimating: the response should match 64 and make sense in the original problem.

Answer: 64

5. Find the discriminant of x^2 - 2x - 15 = 0.

Show solution
  1. Discriminant: First identify exactly what the question is asking: Find the discriminant of x^2 - 2x - 15 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 4 - 4(1)(-15) = 64.
  4. Check the result by substituting or estimating: the response should match 64 and make sense in the original problem.

Answer: 64

Practice this interactively with instant feedback and an AI tutor.

Practice Quadratic Deepening Checkpoint Take the free placement check

More Algebra II lessons