The line y = 2x + b intersects the parabola y = x^2 - 8x + 18 at exactly one...

1. TRANSLATE the problem information

• Given information:
  • Line: \(\mathrm{y = 2x + b}\)
  • Parabola: \(\mathrm{y = x^2 - 8x + 18}\)
  • They intersect at exactly one real solution
  • Need to find: value of b

2. INFER the mathematical condition

• "Exactly one real solution" means the line is tangent to the parabola
• When we set the equations equal, we get a quadratic equation
• For exactly one solution, the discriminant must equal zero

3. Set up the intersection equation

Set the line equal to the parabola:
\(\mathrm{2x + b = x^2 - 8x + 18}\)

4. SIMPLIFY to standard quadratic form

Rearrange everything to one side:
\(\mathrm{0 = x^2 - 8x + 18 - 2x - b}\)
\(\mathrm{0 = x^2 - 10x + (18 - b)}\)

5. INFER which discriminant components to use

• In the form \(\mathrm{ax^2 + bx + c = 0}\), we have:
  • \(\mathrm{a = 1}\)
  • \(\mathrm{b = -10}\) (coefficient of x)
  • \(\mathrm{c = (18 - b)}\) where b is our unknown

6. SIMPLIFY using the discriminant condition

Set discriminant equal to zero:
\(\mathrm{(-10)^2 - 4(1)(18 - b) = 0}\)
\(\mathrm{100 - 4(18 - b) = 0}\)
\(\mathrm{100 - 72 + 4b = 0}\)
\(\mathrm{28 + 4b = 0}\)
\(\mathrm{4b = -28}\)
\(\mathrm{b = -7}\)

Answer: C) -7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one real solution" to the discriminant condition. They might try to solve the system by substitution or just guess, missing that this is specifically about the discriminant being zero.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make sign errors when expanding \(\mathrm{-4(18 - b)}\), getting \(\mathrm{100 - 72 - 4b = 0}\) instead of \(\mathrm{100 - 72 + 4b = 0}\). This gives them \(\mathrm{4b = 28}\), so \(\mathrm{b = 7}\).

This may lead them to select Choice D (7).

The Bottom Line:

The key insight is recognizing that "exactly one solution" translates to a specific mathematical condition (discriminant = 0), not just setting equations equal and solving.