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Quadratic Applications

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

Quadratic models often describe height, area, revenue, and optimization. Zeros can represent times or break-even points, while vertices represent maximum or minimum values.

What you'll learn

Why it matters: Maximum height, maximum revenue, minimum cost, and area optimization are quadratic application signals. The vertex answers the best-value question, while zeros answer when the modeled quantity reaches a target such as ground level or break-even.

Worked example

Problem. A height model is h(t) = -t^2 + 10t. When does the object return to the ground?

  1. Set h(t) = 0.
  2. Factor -t(t - 10) = 0.
  3. The later time is t = 10.

Answer: 10

Practice problems

1. For h(t) = -t^2 + 12t, when does the maximum occur?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For h(t) = -t^2 + 12t, when does the maximum occur?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Halfway between the zeros 0 and 12.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

2. A rectangle has sides x + 3 and x + 8. If x = 2, what is its area?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A rectangle has sides x + 3 and x + 8. If x = 2, what is its area?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The sides are 5 and 10.
  4. Check the result by substituting or estimating: the response should match 50 and make sense in the original problem.

Answer: 50

3. A downward-opening revenue parabola has its maximum at the...

Choices: Vertex · y-intercept only · Left endpoint always · Denominator

Show solution
  1. Challenge: First identify exactly what the question is asking: A downward-opening revenue parabola has its maximum at the...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Opening down means the vertex is highest.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Vertex

4. For h(t) = -t^2 + 6t, when does the maximum occur?

Show solution
  1. Maximums: First identify exactly what the question is asking: For h(t) = -t^2 + 6t, when does the maximum occur?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The axis is halfway between zeros 0 and 6.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. For R(p) = -2p^2 + 40p, what price gives maximum revenue?

Show solution
  1. Revenue: First identify exactly what the question is asking: For R(p) = -2p^2 + 40p, what price gives maximum revenue?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Use -b/(2a): -40/(2 x -2) = 10.
  4. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

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