The area of a square with side length x is 20 less than 5 times the area of a square...

1. TRANSLATE the problem information

• Given information:
  • Area of square with side length x = \(\mathrm{x^2}\)
  • Area of square with side length (x - 3) = \(\mathrm{(x - 3)^2}\)
  • The first area is 20 less than 5 times the second area
• What this tells us: We need to set up equation \(\mathrm{x^2 = 5(x - 3)^2 - 20}\)

2. SIMPLIFY the equation to standard form

• Expand the right side:
  \(\mathrm{x^2 = 5(x^2 - 6x + 9) - 20}\)
  \(\mathrm{x^2 = 5x^2 - 30x + 45 - 20}\)
  \(\mathrm{x^2 = 5x^2 - 30x + 25}\)
• Rearrange to standard form:
  \(\mathrm{x^2 - 5x^2 + 30x - 25 = 0}\)
  \(\mathrm{-4x^2 + 30x - 25 = 0}\)
  \(\mathrm{4x^2 - 30x + 25 = 0}\)

3. INFER the most efficient approach

• The problem asks for the sum of solutions, not individual solutions
• For quadratic \(\mathrm{ax^2 + bx + c = 0}\), sum of roots = \(\mathrm{-b/a}\)
• Here: \(\mathrm{a = 4, b = -30, c = 25}\)

4. SIMPLIFY to get final answer

• Sum of roots = \(\mathrm{-(-30)/4 = 30/4 = 15/2}\)

Answer: \(\mathrm{15/2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "20 less than 5 times" and set up the equation as \(\mathrm{x^2 = 5(x - 3)^2 + 20}\) or \(\mathrm{5(x - 3)^2 = x^2 - 20}\).

This leads to different quadratic equations and incorrect final answers, causing confusion and potentially random guessing.

Second Most Common Error:

Poor INFER reasoning: Students may correctly set up and solve the quadratic but then solve completely for individual roots instead of recognizing they need only the sum. They might get bogged down in the quadratic formula calculations or factoring attempts when the sum of roots formula provides a direct path.

This wastes time and may lead to computational errors in finding individual solutions rather than using the elegant shortcut.

The Bottom Line:

This problem tests whether students can efficiently translate word relationships into equations and then recognize when a shortcut (sum of roots formula) is more appropriate than complete solution. The key insight is that sometimes you don't need to solve completely to answer what's being asked.