In the xy-plane, the line y = 6x + b and the parabola y = -9x^2 + 30x are graphed....

1. TRANSLATE the problem information

• Given information:
  • Line: \(\mathrm{y = 6x + b}\)
  • Parabola: \(\mathrm{y = -9x^2 + 30x}\)
  • They intersect at exactly one point
  • Need to find: value of b

2. INFER the approach

• Key insight: "Exactly one intersection" means the system has exactly one solution
• This happens when we set the equations equal and get a quadratic with discriminant = 0
• Strategy: Set equations equal, rearrange to standard form, use discriminant condition

3. Set equations equal and SIMPLIFY

• Start with: \(\mathrm{6x + b = -9x^2 + 30x}\)
• Rearrange: \(\mathrm{-9x^2 + 30x - 6x - b = 0}\)
• SIMPLIFY to: \(\mathrm{-9x^2 + 24x - b = 0}\)

4. INFER the discriminant condition

• For exactly one solution: \(\mathrm{discriminant = 0}\)
• In our quadratic \(\mathrm{ax^2 + Bx + c = 0}\):
  • \(\mathrm{a = -9, B = 24, c = -b}\)
• Discriminant = \(\mathrm{B^2 - 4ac = 24^2 - 4(-9)(-b)}\)

5. SIMPLIFY the discriminant calculation

• Discriminant = \(\mathrm{576 - 4(-9)(-b) = 576 - 36b}\)
• Set equal to zero: \(\mathrm{576 - 36b = 0}\)
• Solve: \(\mathrm{36b = 576}\), so \(\mathrm{b = 16}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not connecting "exactly one intersection" to \(\mathrm{discriminant = 0}\)

Students might recognize they need to set the equations equal but then try to solve the quadratic directly instead of using the discriminant condition. They might factor or use the quadratic formula, not realizing that the "exactly one intersection" constraint is the key to finding b.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors in discriminant calculation

When calculating \(\mathrm{-4(-9)(-b)}\), students often make sign errors. They might calculate this as \(\mathrm{+36b}\) instead of \(\mathrm{-36b}\), leading to:

\(\mathrm{576 + 36b = 0}\)
\(\mathrm{36b = -576}\)
\(\mathrm{b = -16}\)

This may lead them to select Choice A (-16).

The Bottom Line:

This problem requires recognizing that geometric conditions (exactly one intersection) translate to algebraic conditions (discriminant = 0). Students who treat this as a standard "solve the system" problem miss the critical insight that makes the problem solvable.