y = -2.5y = x^2 + 8x + k In the given system of equations, k is a positive integer...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = -2.5}\)
  • \(\mathrm{y = x^2 + 8x + k}\) (where k is a positive integer)
  • System has no real solutions

• What this tells us: We need to find when this system cannot be solved for any real x-value.

2. INFER the solution approach

• Since both equations equal y, substitution is the natural approach
• This will create a single equation in x that we can analyze
• "No real solutions" means we need to think about when equations have no solutions

3. SIMPLIFY by substitution

• Substitute \(\mathrm{y = -2.5}\) into the second equation:
  \(\mathrm{-2.5 = x^2 + 8x + k}\)

• Rearrange to standard quadratic form:
  \(\mathrm{x^2 + 8x + k + 2.5 = 0}\)

4. INFER when quadratics have no real solutions

• A quadratic \(\mathrm{ax^2 + bx + c = 0}\) has no real solutions when its discriminant is negative
• For our equation: \(\mathrm{a = 1, b = 8, c = k + 2.5}\)
• Discriminant = \(\mathrm{b^2 - 4ac}\)
  \(\mathrm{= 8^2 - 4(1)(k + 2.5)}\)
  \(\mathrm{= 64 - 4k - 10}\)
  \(\mathrm{= 54 - 4k}\)

5. TRANSLATE the condition into an inequality

• For no real solutions: discriminant \(\mathrm{\lt 0}\)
• So: \(\mathrm{54 - 4k \lt 0}\)
• Solving: \(\mathrm{54 \lt 4k}\), which gives us \(\mathrm{13.5 \lt k}\)

6. APPLY CONSTRAINTS to find the answer

• Since k must be a positive integer and \(\mathrm{k \gt 13.5}\)
• The least possible value is \(\mathrm{k = 14}\)

Answer: 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "no real solutions" to discriminant conditions. They might try to solve the system directly by setting the equations equal, get a quadratic, but then attempt to solve it rather than analyze when it has no solutions. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the discriminant but make algebraic errors when simplifying \(\mathrm{64 - 4(k + 2.5)}\) or solving the inequality \(\mathrm{54 - 4k \lt 0}\). They might get \(\mathrm{k \gt 12.5}\) or \(\mathrm{k \gt 15}\), leading them to incorrect values like \(\mathrm{k = 13}\) or \(\mathrm{k = 16}\).

The Bottom Line:

This problem requires recognizing that "no real solutions" is a condition to impose, not a roadblock to solving. The key insight is translating this phrase into the mathematical condition that discriminant \(\mathrm{\lt 0}\).