-{9x^2 + 30x + c = 0} In the given equation, c is a constant. The equation has exactly one...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(-9\mathrm{x}^2 + 30\mathrm{x} + \mathrm{c} = 0\)
  • The equation has exactly one solution
  • We need to find the value of c

2. INFER the key mathematical relationship

• When a quadratic equation has exactly one solution, its discriminant equals zero
• This is our strategy: use the discriminant condition to solve for c

3. TRANSLATE the equation into standard form components

• For the equation \(-9\mathrm{x}^2 + 30\mathrm{x} + \mathrm{c} = 0\):
  • \(\mathrm{a} = -9\) (coefficient of x²)
  • \(\mathrm{b} = 30\) (coefficient of x)
  • \(\mathrm{c} = \mathrm{c}\) (constant term we're finding)

4. SIMPLIFY using the discriminant formula

• Discriminant formula: \(\mathrm{b}^2 - 4\mathrm{ac}\)
• Substitute our values: \((30)^2 - 4(-9)(\mathrm{c})\)
• This gives us: \(900 - 4(-9)\mathrm{c} = 900 + 36\mathrm{c}\)

5. SIMPLIFY the equation when discriminant equals zero

• Set discriminant = 0: \(900 + 36\mathrm{c} = 0\)
• Subtract 900 from both sides: \(36\mathrm{c} = -900\)
• Divide by 36: \(\mathrm{c} = -25\)

Answer: C. -25

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 use other approaches like completing the square without recognizing the discriminant relationship. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the discriminant correctly but make algebraic errors. For example, they might get \(900 - 36\mathrm{c} = 0\) instead of \(900 + 36\mathrm{c} = 0\), leading to \(\mathrm{c} = 25\) instead of \(\mathrm{c} = -25\). Since 25 isn't among the choices, this causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can translate a verbal condition ("exactly one solution") into a mathematical constraint (discriminant = 0). The key insight isn't computational—it's recognizing what "exactly one solution" tells us about the discriminant.