In the xy-plane, the parabola y = x^2 - 6x + 5 and the line y = x + c...

1. TRANSLATE the problem information

• Given information:
  • Parabola: \(\mathrm{y = x^2 - 6x + 5}\)
  • Line: \(\mathrm{y = x + c}\)
  • They intersect at exactly one point
• We need to find the value of c

2. INFER the approach

• To find intersection points, we set the two equations equal
• "Exactly one point" means the resulting quadratic has exactly one solution
• This happens when the discriminant equals zero

3. SIMPLIFY to find the intersection equation

• Set equations equal: \(\mathrm{x^2 - 6x + 5 = x + c}\)
• Rearrange to standard form: \(\mathrm{x^2 - 7x + (5 - c) = 0}\)
• This gives us our quadratic with \(\mathrm{a = 1, b = -7, c = (5 - c)}\)

4. APPLY the discriminant condition

• For exactly one solution: discriminant = 0
• Discriminant = \(\mathrm{b^2 - 4ac}\) = \(\mathrm{(-7)^2 - 4(1)(5 - c)}\)
• SIMPLIFY:
  \(\mathrm{= 49 - 4(5 - c)}\)
  \(\mathrm{= 49 - 20 + 4c}\)
  \(\mathrm{= 29 + 4c}\)

5. SIMPLIFY to solve for c

• Set discriminant equal to zero: \(\mathrm{29 + 4c = 0}\)
• Solve: \(\mathrm{4c = -29}\), so \(\mathrm{c = -29/4}\)

Answer: B) -29/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one intersection point" to the discriminant condition. Instead, they might try to solve the quadratic \(\mathrm{x^2 - 7x + (5 - c) = 0}\) directly using the quadratic formula, not realizing they need to use the constraint about having exactly one solution.

This leads to confusion because they get stuck with an expression involving c rather than a specific value, often causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the discriminant but make arithmetic errors, particularly when expanding \(\mathrm{-4(5 - c) = -20 + 4c}\). Some students incorrectly get \(\mathrm{-20 - 4c}\) instead, leading them to solve \(\mathrm{29 - 4c = 0}\), which gives \(\mathrm{c = 29/4}\).

This may lead them to look for a positive answer among the choices, potentially selecting Choice C (-7) as the closest "reasonable" value.

The Bottom Line:

This problem requires the key insight that geometric constraints (exactly one intersection) translate to algebraic conditions (discriminant = 0). Students who miss this connection get stuck trying to solve an unsolvable problem.