-{2x^2 + 20x + c = 0} In the given equation, c is a constant. The equation has exactly one...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{-2x^2 + 20x + c = 0}\)
  • The equation has exactly one solution
  • c is a constant we need to find

2. INFER the key mathematical condition

• When a quadratic equation has exactly one solution, its discriminant equals zero
• This is our strategic insight: \(\mathrm{discriminant = 0}\) will give us an equation to solve for c

3. TRANSLATE the equation into standard form coefficients

• From \(\mathrm{-2x^2 + 20x + c = 0}\), we identify:
  • \(\mathrm{a = -2}\) (coefficient of x²)
  • \(\mathrm{b = 20}\) (coefficient of x)
  • \(\mathrm{c = c}\) (the constant we're finding)

4. INFER and apply the discriminant condition

• For \(\mathrm{discriminant = 0}\): \(\mathrm{b^2 - 4ac = 0}\)
• Substitute our values: \(\mathrm{(20)^2 - 4(-2)(c) = 0}\)

5. SIMPLIFY to solve for c

• \(\mathrm{(20)^2 - 4(-2)(c) = 0}\)
• \(\mathrm{400 - (-8c) = 0}\)
• \(\mathrm{400 + 8c = 0}\)
• \(\mathrm{8c = -400}\)
• \(\mathrm{c = -50}\)

Answer: B. -50

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't know or remember that "exactly one solution" means \(\mathrm{discriminant = 0}\).

Without this key insight, students may try to solve the equation directly by factoring or using the quadratic formula, but they can't proceed because they don't know the value of c. This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{discriminant = 0}\) but make algebraic errors in the multi-step solution.

For example, they might incorrectly handle the negative signs: \(\mathrm{b^2 - 4ac = 0}\) becomes \(\mathrm{(20)^2 - 4(-2)(c) = 0}\), but they calculate \(\mathrm{400 - 8c = 0}\) instead of \(\mathrm{400 + 8c = 0}\). This leads to \(\mathrm{c = 50}\) instead of \(\mathrm{c = -50}\), but this isn't among the choices, causing confusion.

The Bottom Line:

This problem tests whether students understand the connection between the number of solutions a quadratic has and its discriminant value. The algebra itself is straightforward once you know this relationship.