Question:Consider the system of equations:y = x^2 - 6x + ky = 2x + 1For what value of k does...

1. TRANSLATE the problem information

• Given information:
  • Parabola: \(\mathrm{y = x^2 - 6x + k}\)
  • Line: \(\mathrm{y = 2x + 1}\)
  • Need: Value of k for exactly one solution

2. INFER the mathematical approach

• For exactly one solution, the parabola and line must intersect at exactly one point
• This happens when setting the equations equal gives a quadratic with exactly one solution
• A quadratic has exactly one solution when its discriminant equals zero

3. Set the equations equal to find intersection condition

Since both expressions equal y:

\(\mathrm{x^2 - 6x + k = 2x + 1}\)

4. SIMPLIFY to standard quadratic form

Move all terms to one side:

\(\mathrm{x^2 - 6x + k - 2x - 1 = 0}\)

\(\mathrm{x^2 - 8x + (k - 1) = 0}\)

5. APPLY CONSTRAINTS using discriminant condition

For exactly one solution, discriminant = 0

Using \(\mathrm{b^2 - 4ac}\) where \(\mathrm{a = 1, b = -8, c = (k - 1)}\):

Discriminant = \(\mathrm{(-8)^2 - 4(1)(k - 1) = 64 - 4(k - 1) = 68 - 4k}\)

6. SIMPLIFY the final equation

Set discriminant equal to zero:

\(\mathrm{68 - 4k = 0}\)

\(\mathrm{4k = 68}\)

\(\mathrm{k = 17}\)

Answer: 17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one solution" with the discriminant condition. Instead, they might try to solve the system directly by substitution or think they need to find the actual solution point rather than the parameter value k.

This leads to confusion about what they're actually solving for, causing them to get stuck and abandon systematic solution.

Second Most Common Error:

Calculation errors in SIMPLIFY: Students correctly set up the discriminant equation but make arithmetic mistakes when expanding \(\mathrm{4(k - 1)}\) or combining terms, leading to expressions like \(\mathrm{64 - 4k - 4}\) instead of \(\mathrm{64 - 4k + 4}\).

This leads to incorrect values like \(\mathrm{k = 15}\) instead of \(\mathrm{k = 17}\).

The Bottom Line:

This problem requires students to think conceptually about what "exactly one solution" means for intersections, then execute the discriminant calculation accurately. The key insight is recognizing that intersection conditions translate to discriminant conditions.