Question:Two numbers have a sum of -{12} and a product of -{9}. Which of the following could be the value...

1. INFER the mathematical relationship

• Given information:
  • Two numbers sum to -12
  • Two numbers have product of -9

• Key insight: When you know both the sum and product of two numbers, those numbers are the roots of a quadratic equation!

2. INFER the quadratic equation form

• For any quadratic with roots having sum S and product P:
  The equation is \(\mathrm{x^2 - Sx + P = 0}\)

• Substituting our values (\(\mathrm{S = -12, P = -9}\)):
  \(\mathrm{x^2 - (-12)x + (-9) = 0}\)
  \(\mathrm{x^2 + 12x - 9 = 0}\)

3. SIMPLIFY using the quadratic formula

• Apply \(\mathrm{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\) where \(\mathrm{a = 1, b = 12, c = -9}\)

• \(\mathrm{x = \frac{-12 \pm \sqrt{12^2 - 4(1)(-9)}}{2 \times 1}}\)

• \(\mathrm{x = \frac{-12 \pm \sqrt{144 + 36}}{2}}\)

• \(\mathrm{x = \frac{-12 \pm \sqrt{180}}{2}}\)

4. SIMPLIFY the radical

• Factor 180 to find perfect squares:
  \(\mathrm{180 = 36 \times 5 = 6^2 \times 5}\)

• Therefore: \(\mathrm{\sqrt{180} = 6\sqrt{5}}\)

• \(\mathrm{x = \frac{-12 \pm 6\sqrt{5}}{2} = -6 \pm 3\sqrt{5}}\)

5. Identify the answer

• The two solutions are: \(\mathrm{-6 + 3\sqrt{5}}\) and \(\mathrm{-6 - 3\sqrt{5}}\)

• Looking at the choices, \(\mathrm{-6 + 3\sqrt{5}}\) appears as option (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between "sum and product" and quadratic equations. Instead, they try to set up a system of equations like \(\mathrm{x + y = -12}\) and \(\mathrm{xy = -9}\), then struggle to solve it algebraically without realizing this leads to the same quadratic.

This leads to confusion and guessing, or attempting to substitute answer choices to check which works.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the quadratic but make errors when applying the quadratic formula, particularly with the discriminant calculation (\(\mathrm{12^2 - 4(1)(-9) = 144 + 36}\), not \(\mathrm{144 - 36}\)) or when simplifying \(\mathrm{\sqrt{180}}\).

Common mistakes include getting \(\mathrm{\sqrt{180} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}}\) wrong, or making sign errors in the final expression.

This may lead them to select Choice A (\(\mathrm{6 - 3\sqrt{5}}\)) if they get the wrong signs.

The Bottom Line:

This problem tests whether students can connect the abstract concept of "sum and product of two numbers" to the concrete tool of quadratic equations. Once that connection is made, it becomes a straightforward quadratic formula problem.