\(3(\mathrm{p}^2 + 3) = -12\mathrm{p}\)What is the sum of the solutions to the given equation?

1. SIMPLIFY the equation to standard form

• Start with: \(3(\mathrm{p}^2 + 3) = -12\mathrm{p}\)

• SIMPLIFY by distributing the 3 on the left side:
  • \(3(\mathrm{p}^2 + 3) = 3\mathrm{p}^2 + 9\)
  • Equation becomes: \(3\mathrm{p}^2 + 9 = -12\mathrm{p}\)

• SIMPLIFY further by moving all terms to one side:
  • Add 12p to both sides: \(3\mathrm{p}^2 + 12\mathrm{p} + 9 = 0\)

2. INFER the most efficient solution strategy

• Now we have standard form: \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\) where \(\mathrm{a} = 3, \mathrm{b} = 12, \mathrm{c} = 9\)

• INFER that we can use the sum of roots formula instead of finding individual roots:
  • Sum of solutions = \(-\mathrm{b}/\mathrm{a}\)

3. SIMPLIFY using the formula

• SIMPLIFY the calculation:
  • Sum = \(-12/3 = -4\)

Answer: -4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when rearranging terms to standard form.

A typical mistake is writing \(3\mathrm{p}^2 + 9 = -12\mathrm{p}\) as \(3\mathrm{p}^2 - 12\mathrm{p} + 9 = 0\) (forgetting that -12p becomes +12p when moved to the left side). This gives incorrect coefficients and leads to the wrong sum calculation.

This may lead them to calculate sum as \(-(-12)/3 = +4\) instead of -4.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember the sum of roots formula.

Without knowing that \(\text{sum} = -\mathrm{b}/\mathrm{a}\), students attempt to factor and solve for individual roots but make errors in the factoring process. They might incorrectly factor \(\mathrm{p}^2 + 4\mathrm{p} + 3\) or make arithmetic mistakes when adding the final solutions.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students can systematically rearrange equations and apply quadratic relationships efficiently. The key insight is recognizing that finding the sum directly is much faster than finding individual solutions.