The product of two positive integers is 462. If the first integer is 5 greater than twice the second integer,...

1. TRANSLATE the problem information

• Given information:
  • Two positive integers have a product of 462
  • First integer = 5 greater than twice the second integer
  • Need to find the smaller integer

• Let \(\mathrm{y}\) = second integer, then first integer = \(\mathrm{2y + 5}\)

2. TRANSLATE the constraint into an equation

• Since the product equals 462:
  \(\mathrm{y(2y + 5) = 462}\)

3. SIMPLIFY by expanding and rearranging

• Expand: \(\mathrm{2y^2 + 5y = 462}\)
• Rearrange: \(\mathrm{2y^2 + 5y - 462 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to \(\mathrm{(2)(-462) = -924}\) and add to 5
• Those numbers are 33 and -28: \(\mathrm{(33)(-28) = -924}\), \(\mathrm{33 + (-28) = 5}\)
• Rewrite: \(\mathrm{2y^2 + 33y - 28y - 462 = 0}\)
• Factor by grouping: \(\mathrm{y(2y + 33) - 14(2y + 33) = 0}\)
• Factor completely: \(\mathrm{(y - 14)(2y + 33) = 0}\)

5. APPLY CONSTRAINTS to select valid solutions

• From zero product property: \(\mathrm{y = 14}\) or \(\mathrm{y = -33/2}\)
• Since we need positive integers: \(\mathrm{y = 14}\)
• Therefore: first integer = \(\mathrm{2(14) + 5 = 33}\)
• The two integers are 14 and 33

Answer: 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to correctly set up the relationship '5 greater than twice the second integer.' They might write the first integer as \(\mathrm{2y - 5}\) or \(\mathrm{5 + y}\) instead of \(\mathrm{2y + 5}\). This leads to an incorrect quadratic equation and completely wrong solutions. This causes confusion and leads to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students get the setup correct but make errors in factoring the quadratic \(\mathrm{2y^2 + 5y - 462 = 0}\). They might struggle to find the correct factor pair \(\mathrm{(33, -28)}\) or make algebraic mistakes during the factoring process. This leads to incorrect or incomplete solutions and causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires precise translation of verbal relationships into algebraic expressions, followed by systematic quadratic solving. Students who rush through the initial setup or lack confidence in factoring quadratics will struggle to reach the correct answer.