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

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x^2 + 6x + 7 = 0}\) has solutions r and s (\(\mathrm{r \lt s}\))
  • \(\mathrm{x^2 + 8x + 8 = 0}\) has solutions t and u (\(\mathrm{t \lt u}\))
  • \(\mathrm{x^2 + 14x + c = 0}\) has solutions \(\mathrm{r + t}\) and \(\mathrm{s + u}\)
  • Need to find the constant c

2. INFER the solution strategy

• Key insight: We need to find the individual roots first, then use them to determine c
• Since we know the solutions to the third equation are \(\mathrm{r + t}\) and \(\mathrm{s + u}\), we can use Vieta's formulas
• For \(\mathrm{x^2 + 14x + c = 0}\), the product of roots equals c

3. SIMPLIFY to find the first set of roots

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

4. SIMPLIFY to find the second set of roots

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

5. SIMPLIFY to find the new roots

• \(\mathrm{r + t = (-3 - \sqrt{2}) + (-4 - 2\sqrt{2}) = -7 - 3\sqrt{2}}\)
• \(\mathrm{s + u = (-3 + \sqrt{2}) + (-4 + 2\sqrt{2}) = -7 + 3\sqrt{2}}\)

6. INFER how to find c using Vieta's formulas

• For \(\mathrm{x^2 + 14x + c = 0}\), the product of roots equals c
• \(\mathrm{c = (r + t)(s + u) = (-7 - 3\sqrt{2})(-7 + 3\sqrt{2})}\)

7. SIMPLIFY using difference of squares

• \(\mathrm{(-7 - 3\sqrt{2})(-7 + 3\sqrt{2}) = (-7)^2 - (3\sqrt{2})^2}\)
  \(\mathrm{= 49 - 9(2)}\)
  \(\mathrm{= 49 - 18}\)
  \(\mathrm{= 31}\)

Answer: 31

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to find individual roots first, then use Vieta's formulas to find c. Instead, they might try to work directly with the coefficients or attempt complex substitutions. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when computing \(\mathrm{(3\sqrt{2})^2 = 18}\), incorrectly calculating it as 6 instead of 18. This would give them \(\mathrm{(-7)^2 - 6 = 49 - 6 = 43}\) instead of the correct answer. This may lead them to select an incorrect answer choice if 43 were available.

The Bottom Line:

This problem requires recognizing the connection between roots of different quadratics through Vieta's formulas, combined with careful radical arithmetic. The key insight is that finding c requires computing the product of the new roots, not trying to relate the coefficients directly.