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Example

Finding the Height and Width of a Triangular Window with Area 120 Square Feet

Apply the seven-step problem-solving strategy to find the dimensions of a triangle when the area and a relationship between the width and height are known, producing a quadratic equation solved using the Quadratic Formula.

Problem: An architect wants a triangular window with an area of 120120 square feet and a width that is 44 feet more than twice the height. Find the height and width of the window.

  1. Read: A triangular window has A=120A = 120 sq ft, and its width is 44 more than twice its height. Draw and label the triangle.
  2. Identify: The height and width of the triangle.
  3. Name: Let hh = the height of the triangle. Then 2h+42h + 4 = the width of the triangle.
  4. Translate: Write the triangle area formula and substitute:

A=12bhA = \frac{1}{2}bh

120=12(2h+4)(h)120 = \frac{1}{2}(2h + 4)(h)

  1. Solve: Distribute 12\frac{1}{2} and hh on the right side: 120=h2+2h120 = h^2 + 2h. Rewrite in standard form: h2+2h−120=0h^2 + 2h - 120 = 0. Identify coefficients: a=1a = 1, b=2b = 2, c=−120c = -120. Substitute into the Quadratic Formula:

h=−2±22−4(1)(−120)2(1)=−2±4+4802=−2±4842h = \frac{-2 \pm \sqrt{2^2 - 4(1)(-120)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 480}}{2} = \frac{-2 \pm \sqrt{484}}{2}

Since 484=22\sqrt{484} = 22:

h=−2+222=10orh=−2−222=−12h = \frac{-2 + 22}{2} = 10 \quad \text{or} \quad h = \frac{-2 - 22}{2} = -12

Because hh represents a physical height, h=−12h = -12 is discarded. So h=10h = 10, and the width is 2(10)+4=242(10) + 4 = 24.

  1. Check: A=12(24)(10)=120A = \frac{1}{2}(24)(10) = 120 sq ft ✓
  2. Answer: The height of the triangular window is 1010 feet and the width is 2424 feet.

Because the discriminant 484484 turned out to be a perfect square, the solutions were integers. This means the equation h2+2h−120=0h^2 + 2h - 120 = 0 could also have been solved by factoring: (h+12)(h−10)=0(h + 12)(h - 10) = 0. When the Quadratic Formula produces integer or rational solutions, factoring would have been an equally valid — and often faster — alternative.

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Updated 2026-04-21

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References

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OpenStax

Elementary Algebra @ OpenStax

Ch.10 Quadratic Equations - Elementary Algebra @ OpenStax

Algebra

Math

Prealgebra

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