The quadratic equation x^2 - 7x + k = 0 has roots 3 and 4. A new quadratic equation has...

1. TRANSLATE the problem information

• Given information:
  • Original equation: \(\mathrm{x^2 - 7x + k = 0}\) with roots 3 and 4
  • New equation has roots that are 2 times each original root
  • New equation form: \(\mathrm{x^2 - 14x + m = 0}\)

2. INFER the connection between roots and coefficients

• We can use Vieta's formulas to connect roots to equation coefficients
• For quadratic \(\mathrm{x^2 + bx + c = 0}\): sum of roots = -b, product of roots = c
• This lets us work with the roots directly rather than solving equations

3. Verify the original equation setup

• Check that roots 3 and 4 work with \(\mathrm{x^2 - 7x + k = 0}\):
  • Sum: \(\mathrm{3 + 4 = 7}\) (matches \(\mathrm{-(-7)}\))
  • Product: \(\mathrm{3 \times 4 = 12}\), so \(\mathrm{k = 12}\) ✓

4. TRANSLATE to find the new roots

• New roots are twice the original roots:
  • First new root: \(\mathrm{2 \times 3 = 6}\)
  • Second new root: \(\mathrm{2 \times 4 = 8}\)

5. SIMPLIFY to find the constant term m

• For equation \(\mathrm{x^2 - 14x + m = 0}\) with roots 6 and 8:
  • Sum check: \(\mathrm{6 + 8 = 14}\) ✓ (matches coefficient pattern)
  • Product: \(\mathrm{m = 6 \times 8 = 48}\)

Answer: 48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the power of Vieta's formulas to directly connect roots and coefficients.

Students often try to construct the new quadratic by expanding \(\mathrm{(x - 6)(x - 8)}\), which works but takes longer and creates more opportunities for algebraic errors. Some students get lost trying to find the original value of k first, not realizing they can work directly with the given roots.

This leads to confusion and potentially abandoning the systematic approach in favor of guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when calculating the product of new roots.

Students correctly identify that the new roots are 6 and 8, but then miscalculate \(\mathrm{6 \times 8}\), perhaps getting 46 or 44 instead of 48. Since this is the final step, any arithmetic error here directly leads to the wrong answer.

The Bottom Line:

This problem rewards students who recognize that Vieta's formulas provide the most direct path from roots to coefficients, avoiding unnecessary algebraic manipulation while requiring careful arithmetic in the final step.