6x^2 + 5x - 7 = 0What are the solutions to the given equation?

1. TRANSLATE the problem information

• Given equation: \(\mathrm{6x^2 + 5x - 7 = 0}\)
• This is in standard form \(\mathrm{ax^2 + bx + c = 0}\), so we can identify:
  • \(\mathrm{a = 6}\) (coefficient of \(\mathrm{x^2}\))
  • \(\mathrm{b = 5}\) (coefficient of \(\mathrm{x}\))
  • \(\mathrm{c = -7}\) (constant term)

2. INFER the approach

• Since we have a quadratic equation, we'll use the quadratic formula
• The quadratic formula is: \(\mathrm{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\)
• We need to substitute our values for a, b, and c

3. SIMPLIFY by substituting into the quadratic formula

• \(\mathrm{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\)
• \(\mathrm{x = \frac{-5 \pm \sqrt{5^2 - 4(6)(-7)}}{2(6)}}\)
• \(\mathrm{x = \frac{-5 \pm \sqrt{25 - 4(6)(-7)}}{12}}\)

4. SIMPLIFY the discriminant (the part under the square root)

• \(\mathrm{b^2 - 4ac = 5^2 - 4(6)(-7)}\)
• \(\mathrm{= 25 - (-168)}\)
• \(\mathrm{= 25 + 168}\)

5. Write the final form

• \(\mathrm{x = \frac{-5 \pm \sqrt{25 + 168}}{12}}\)

Answer: A. \(\mathrm{\frac{-5 \pm \sqrt{25 + 168}}{12}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing which coefficient goes where in the quadratic formula

Students often mix up -b with -a, thinking they need to use the negative of the \(\mathrm{x^2}\) coefficient instead of the negative of the x coefficient. Since \(\mathrm{a = 6}\), they write \(\mathrm{(-6 \pm ...)}\) instead of \(\mathrm{(-5 \pm ...)}\).

This leads them to select Choice B (\(\mathrm{\frac{-6 \pm \sqrt{25 + 168}}{12}}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Incorrectly identifying what goes into the discriminant

Students might substitute \(\mathrm{a^2}\) instead of \(\mathrm{b^2}\) into the discriminant, getting \(\mathrm{\sqrt{36 - 4ac}}\) instead of \(\mathrm{\sqrt{25 - 4ac}}\). They may also make sign errors with 4ac, writing it as +4ac instead of -4ac when c is negative.

This leads them to select Choice C (\(\mathrm{\frac{-5 \pm \sqrt{36 - 168}}{12}}\)) or Choice D (\(\mathrm{\frac{-6 \pm \sqrt{36 - 168}}{12}}\))

The Bottom Line:

The quadratic formula has many moving parts, and students need to be methodical about identifying each coefficient correctly and placing it in the right position in the formula. Small errors in coefficient identification cascade into completely wrong answer choices.