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MathidiaMathematics Academy
Quadratic equations and graphs
Lesson 20 of 75
Review

Quadratic modelling

Objective. Form and solve quadratic equations from area and motion contexts.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Quadratic modelling involves translating a real situation, such as areas or projectile motion, into a quadratic equation and then solving it to answer the original question. Common contexts include a rectangle's area expressed in terms of one unknown side, or the height of an object given by a formula in time t. After forming the equation and solving it, always check each solution against the context, since negative lengths or times are usually invalid even if they are valid algebraic solutions. State the final answer with correct units.

Worked example

A rectangular garden has length (x + 4) m and width x m, with area 32 m². Form and solve a quadratic equation to find x.

Diagram of a rectangle
  1. 1Form the equation from the area: x(x + 4) = 32.
  2. 2Expand and rearrange: x² + 4x − 32 = 0.
  3. 3Factorise: (x + 8)(x − 4) = 0, giving x = −8 or x = 4.
  4. 4Reject the negative length: x = 4 m.

Answer: x = 4 m

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Practice

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Common mistake. Pupils give both algebraic solutions of a quadratic as final answers even when one is physically impossible, such as a negative length. Always discard solutions that do not make sense in context.

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