Question:4c^2 - c - 3 = 0Let s be the negative solution to the equation above. What is the value...

1. TRANSLATE the problem information

• Given information:
  • Quadratic equation: \(\mathrm{4c^2 - c - 3 = 0}\)
  • s represents the negative solution
  • Need to find the value of -s

2. INFER the approach

• Since we have a quadratic equation, we need to find both solutions using the quadratic formula
• Then identify which solution is negative (that will be s)
• Finally calculate -s from that negative solution

3. SIMPLIFY using the quadratic formula

• Identify coefficients: \(\mathrm{a = 4, b = -1, c = -3}\)
• Calculate discriminant: \(\mathrm{b^2 - 4ac = (-1)^2 - 4(4)(-3) = 1 + 48 = 49}\)
• Apply formula: \(\mathrm{c = \frac{1 ± \sqrt{49}}{8} = \frac{1 ± 7}{8}}\)

4. SIMPLIFY to find both solutions

• First solution: \(\mathrm{c = \frac{1 + 7}{8} = \frac{8}{8} = 1}\)
• Second solution: \(\mathrm{c = \frac{1 - 7}{8} = \frac{-6}{8} = -\frac{3}{4}}\)

5. APPLY CONSTRAINTS to identify the negative solution

• Since s is defined as the negative solution: \(\mathrm{s = -\frac{3}{4}}\)
• The problem asks for -s: \(\mathrm{-s = -(-\frac{3}{4}) = \frac{3}{4}}\)

Answer: 3/4 (or 0.75)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students solve for both solutions correctly but then provide \(\mathrm{s = -\frac{3}{4}}\) as their final answer instead of recognizing they need to find -s.

They correctly identify that \(\mathrm{s = -\frac{3}{4}}\), but miss the crucial final step where the problem asks for the value of -s, not s itself. This leads them to answer -3/4 or -0.75 instead of the correct 3/4.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when applying the quadratic formula, particularly with the negative coefficient \(\mathrm{b = -1}\).

They might write the formula as \(\mathrm{c = \frac{-(-1) ± \sqrt{49}}{8}}\) but then incorrectly simplify this to \(\mathrm{c = \frac{-1 ± 7}{8}}\) instead of \(\mathrm{c = \frac{1 ± 7}{8}}\). This leads to wrong solutions and cascades through the rest of the problem.

The Bottom Line:

This problem tests both computational accuracy with the quadratic formula and careful attention to what the problem is actually asking for. Many students can solve quadratic equations but stumble on the interpretation step where they need -s rather than s.