The product of a positive number x and the number that is 8 more than x is 180. What is...

1. TRANSLATE the problem information

• Given information:
  • A positive number: \(\mathrm{x}\) (where \(\mathrm{x \gt 0}\))
  • The number that is 8 more than \(\mathrm{x}\): \(\mathrm{x + 8}\)
  • Their product equals 180
• This tells us: \(\mathrm{x(x + 8) = 180}\)

2. SIMPLIFY to standard form

• Expand the left side: \(\mathrm{x^2 + 8x = 180}\)
• Move all terms to one side: \(\mathrm{x^2 + 8x - 180 = 0}\)
• Now we have a quadratic equation in standard form

3. SIMPLIFY by factoring

• We need two numbers that multiply to \(\mathrm{-180}\) and add to 8
• Testing factors of 180: We find 18 and -10 work perfectly
  • \(\mathrm{18 × (-10) = -180}\) ✓
  • \(\mathrm{18 + (-10) = 8}\) ✓
• Factor: \(\mathrm{(x + 18)(x - 10) = 0}\)

4. INFER the solutions using zero product property

• If \(\mathrm{(x + 18)(x - 10) = 0}\), then either:
  • \(\mathrm{x + 18 = 0}\), so \(\mathrm{x = -18}\), OR
  • \(\mathrm{x - 10 = 0}\), so \(\mathrm{x = 10}\)

5. APPLY CONSTRAINTS to select the valid answer

• The problem states \(\mathrm{x}\) is positive
• Therefore, \(\mathrm{x = -18}\) is invalid
• The answer is \(\mathrm{x = 10}\)

Answer: B. 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread "8 more than x" as "8 less than x"

Instead of setting up \(\mathrm{x(x + 8) = 180}\), they write \(\mathrm{x(x - 8) = 180}\). This leads to \(\mathrm{x^2 - 8x - 180 = 0}\), which factors as \(\mathrm{(x - 18)(x + 10) = 0}\). Applying the positive constraint gives \(\mathrm{x = 18}\).

This may lead them to select Choice C (18).

The Bottom Line:

This problem tests your ability to carefully translate English into mathematical expressions. The phrase "8 more than x" specifically means "\(\mathrm{x + 8}\)," not "\(\mathrm{x - 8}\)." Once translated correctly, the algebraic steps follow standard quadratic-solving procedures, but that initial translation step is crucial for success.