-{16x^2 - 8x + c = 0} In the given equation, c is a constant. The equation has exactly one...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(-16\mathrm{x}^2 - 8\mathrm{x} + \mathrm{c} = 0\)
  • The equation has exactly one solution
  • Need to find the value of c

2. INFER the mathematical condition

• "Exactly one solution" means the discriminant must equal zero
• For any quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), the discriminant is \(\mathrm{b}^2 - 4\mathrm{ac}\)
• When discriminant = 0, the quadratic has exactly one solution

3. TRANSLATE the equation to identify coefficients

• From \(-16\mathrm{x}^2 - 8\mathrm{x} + \mathrm{c} = 0\):
  • \(\mathrm{a} = -16\) (coefficient of x²)
  • \(\mathrm{b} = -8\) (coefficient of x)
  • \(\mathrm{c} = \mathrm{c}\) (the constant we're solving for)

4. SIMPLIFY by setting up the discriminant equation

• Discriminant = \(\mathrm{b}^2 - 4\mathrm{ac} = 0\)
• Substitute: \((-8)^2 - 4(-16)(\mathrm{c}) = 0\)
• Calculate: \(64 - 4(-16)(\mathrm{c}) = 0\)
• Simplify: \(64 + 64\mathrm{c} = 0\)

5. SIMPLIFY to solve for c

• \(64 + 64\mathrm{c} = 0\)
• \(64\mathrm{c} = -64\)
• \(\mathrm{c} = -1\)

Answer: -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one solution" to discriminant = 0. They might try to solve the quadratic directly or guess values for c without understanding the underlying mathematical condition.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that discriminant = 0 is needed, but make sign errors when identifying coefficients, especially with \(\mathrm{a} = -16\). They might use \(\mathrm{a} = 16\) instead.

This leads to incorrect setup: \((-8)^2 - 4(16)(\mathrm{c}) = 0\), which gives \(64 - 64\mathrm{c} = 0\), resulting in \(\mathrm{c} = 1\) instead of \(\mathrm{c} = -1\).

The Bottom Line:

This problem requires connecting the phrase "exactly one solution" to a specific algebraic condition (discriminant = 0). Students who memorize discriminant formulas but don't understand what different discriminant values mean will struggle to even begin this problem systematically.