The solutions to the equation x^2 + 6x + 7 = 0 are r and s, where r lt s....

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{x^2 + 6x + 7 = 0}\) has roots r and s \(\mathrm{(r \lt s)}\)
  • Second equation: \(\mathrm{x^2 + 8x + 8 = 0}\) has roots t and u \(\mathrm{(t \lt u)}\)
  • Third equation: \(\mathrm{x^2 + bx + 56 = 0}\) has roots rt and su
  • Need to find: the value of b

2. INFER the solution strategy

• We need the roots r, s, t, u first to calculate the products rt and su
• Once we have rt and su, we can use Vieta's formulas on the third equation to find b
• The key insight: the sum of roots \(\mathrm{rt + su = -b}\) for the equation \(\mathrm{x^2 + bx + 56 = 0}\)

3. SIMPLIFY to find the roots of the first equation

• Apply quadratic formula to \(\mathrm{x^2 + 6x + 7 = 0}\):
  \(\mathrm{x = \frac{-6 \pm \sqrt{36-28}}{2}}\)
  \(\mathrm{= \frac{-6 \pm \sqrt{8}}{2}}\)
  \(\mathrm{= \frac{-6 \pm 2\sqrt{2}}{2}}\)
  \(\mathrm{= -3 \pm \sqrt{2}}\)
• Since \(\mathrm{r \lt s}\): \(\mathrm{r = -3 - \sqrt{2}}\) and \(\mathrm{s = -3 + \sqrt{2}}\)

4. SIMPLIFY to find the roots of the second equation

• Apply quadratic formula to \(\mathrm{x^2 + 8x + 8 = 0}\):
  \(\mathrm{x = \frac{-8 \pm \sqrt{64-32}}{2}}\)
  \(\mathrm{= \frac{-8 \pm \sqrt{32}}{2}}\)
  \(\mathrm{= \frac{-8 \pm 4\sqrt{2}}{2}}\)
  \(\mathrm{= -4 \pm 2\sqrt{2}}\)
• Since \(\mathrm{t \lt u}\): \(\mathrm{t = -4 - 2\sqrt{2}}\) and \(\mathrm{u = -4 + 2\sqrt{2}}\)

5. SIMPLIFY to calculate the new roots rt and su

• \(\mathrm{rt = (-3 - \sqrt{2})(-4 - 2\sqrt{2})}\)
  Using FOIL: \(\mathrm{rt = 12 + 6\sqrt{2} + 4\sqrt{2} + 4}\)
  \(\mathrm{= 16 + 10\sqrt{2}}\)
• \(\mathrm{su = (-3 + \sqrt{2})(-4 + 2\sqrt{2})}\)
  Using FOIL: \(\mathrm{su = 12 - 6\sqrt{2} - 4\sqrt{2} + 4}\)
  \(\mathrm{= 16 - 10\sqrt{2}}\)

6. INFER using Vieta's formulas

• For equation \(\mathrm{x^2 + bx + 56 = 0}\), the sum of roots equals -b
• Sum = \(\mathrm{rt + su}\)
  \(\mathrm{= (16 + 10\sqrt{2}) + (16 - 10\sqrt{2})}\)
  \(\mathrm{= 32}\)
• Therefore: \(\mathrm{-b = 32}\), which means \(\mathrm{b = -32}\)

Answer: A) -32

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when multiplying the binomials containing radicals, especially when calculating \(\mathrm{rt = (-3 - \sqrt{2})(-4 - 2\sqrt{2})}\) and \(\mathrm{su = (-3 + \sqrt{2})(-4 + 2\sqrt{2})}\). They might incorrectly handle the radical terms or make sign errors during FOIL expansion.

For example, they might calculate rt as \(\mathrm{16 + 6\sqrt{2}}\) (missing the \(\mathrm{4\sqrt{2}}\) term) or get the wrong signs. This leads to an incorrect sum rt + su, giving them a wrong value for b. This may lead them to select Choice B (-16) or Choice D (32).

Second Most Common Error:

Poor INFER reasoning: Students might not recognize the connection between Vieta's formulas and finding b. They calculate rt and su correctly but then don't know how to use these products as roots of the third equation. They might try to use the product of roots (which equals 56) instead of the sum of roots (which equals -b).

This leads to confusion about how to connect their calculated values to the coefficient b, causing them to get stuck and guess.

The Bottom Line:

This problem requires strong algebraic manipulation skills with radicals AND the strategic insight to connect Vieta's formulas across multiple equations. Students who struggle with either aspect will have difficulty reaching the correct answer.