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Quadratic Applications

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

Quadratics appear in area, projectile motion, revenue, and other situations with changing rates. The key is deciding whether the question asks for a zero, vertex, or value.

What you'll learn

Why it matters: Quadratic applications connect symbols to gardens, packaging, ball flight, and revenue curves where zeros and vertices carry the meaning.

Worked example

Problem. A ball has height h = -t^2 + 6t. When does it hit the ground?

  1. The ball hits the ground when h = 0.
  2. Set -t^2 + 6t = 0 and factor: -t(t - 6) = 0.
  3. t = 0 is the launch time, so it hits the ground at t = 6.

Answer: 6 seconds

Practice problems

1. A rectangle has area x(x + 5). If x = 4, what is the area?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A rectangle has area x(x + 5). If x = 4, what is the area?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Substitute x = 4.
  4. 4(9) = 36.
  5. Check the result by substituting or estimating: the response should match 36 and make sense in the original problem.

Answer: 36

2. A ball has height h = -t^2 + 4t. At t = 2, what is h?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A ball has height h = -t^2 + 4t. At t = 2, what is h?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute t = 2.
  4. -4 + 8 = 4.
  5. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. If a height model equals 0, the object is...

Choices: At ground level · At maximum height · Moving fastest · At the y-intercept only

Show solution
  1. Warm-up: First identify exactly what the question is asking: If a height model equals 0, the object is...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Height zero means ground level.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: At ground level

4. A ball has height h = -t^2 + 5t. Other than t = 0, when is h = 0?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A ball has height h = -t^2 + 5t. Other than t = 0, when is h = 0?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor -t(t - 5) = 0.
  4. The nonzero time is 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. A rectangle has sides x + 2 and x + 6. If x = 3, what is the area?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A rectangle has sides x + 2 and x + 6. If x = 3, what is the area?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The side lengths are 5 and 9.
  4. 5 x 9 = 45.
  5. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

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