Quadratic Foundations Checkpoint
A free Algebra I lesson from the “Quadratic Foundations” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint pulls together factoring, solving, graphing, and applications of quadratics. It asks whether you can connect symbolic form to features of a parabola and to real contexts.
What you'll learn
- Factor and solve quadratics
- Interpret parabolas, vertices, and zeros
- Apply quadratic models in context
Worked example
Problem. Solve x^2 - 9x + 20 = 0 and name the zeros.
- Factor x^2 - 9x + 20 as (x - 4)(x - 5).
- Set each factor equal to zero.
- The zeros are 4 and 5.
Answer: x = 4 and x = 5
Practice problems
1. Factor x^2 + 9x + 20.
Choices: (x + 4)(x + 5) · (x + 2)(x + 10) · (x - 4)(x - 5) · (x + 1)(x + 20)
Show solution
- Factoring: First identify exactly what the question is asking: Factor x^2 + 9x + 20.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 4 and 5 multiply to 20 and add to 9.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (x + 4)(x + 5)
2. Factor x^2 - 5x - 14.
Choices: (x - 7)(x + 2) · (x + 7)(x - 2) · (x - 14)(x + 1) · (x - 5)(x + 14)
Show solution
- Factoring: First identify exactly what the question is asking: Factor x^2 - 5x - 14.
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- -7 and 2 multiply to -14 and add to -5.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: (x - 7)(x + 2)
3. Solve (x - 3)(x + 8) = 0.
Choices: 3 and -8 · -3 and 8 · 3 and 8 · -3 and -8
Show solution
- Solving: First identify exactly what the question is asking: Solve (x - 3)(x + 8) = 0.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Set each factor equal to zero.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 3 and -8
4. Solve x^2 - 7x + 10 = 0.
Choices: 2 and 5 · -2 and -5 · 1 and 10 · 3 and 4
Show solution
- Solving: First identify exactly what the question is asking: Solve x^2 - 7x + 10 = 0.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Factor as (x - 2)(x - 5).
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 2 and 5
5. Solve x^2 - 49 = 0. Enter the positive solution.
Show solution
- Solving: First identify exactly what the question is asking: Solve x^2 - 49 = 0. Enter the positive solution.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Factor as (x - 7)(x + 7).
- Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.
Answer: 7
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