sqrt(k) - x = 58 - x In the given equation, k is a constant. The equation has exactly one...

1. TRANSLATE the problem information

• Given equation: \(\sqrt{\mathrm{k - x}} = 58 - \mathrm{x}\)
• Constraint: The equation has exactly one real solution
• Find: Minimum value of 4k

2. INFER the domain restrictions

• For \(\sqrt{\mathrm{k - x}}\) to be defined, we need \(\mathrm{k - x} \geq 0\)
• For the equation to be valid, we need \(58 - \mathrm{x} \geq 0\) (since square root equals this)
• This means \(\mathrm{x} \leq 58\)

3. SIMPLIFY by eliminating the radical

• Square both sides: \((\sqrt{\mathrm{k - x}})^2 = (58 - \mathrm{x})^2\)
• Left side: \(\mathrm{k - x}\)
• Right side: \((58 - \mathrm{x})^2 = 58^2 - 2(58)\mathrm{x} + \mathrm{x}^2 = 3364 - 116\mathrm{x} + \mathrm{x}^2\)
• Result: \(\mathrm{k - x} = 3364 - 116\mathrm{x} + \mathrm{x}^2\)

4. SIMPLIFY to standard quadratic form

• Rearrange: \(\mathrm{k} = \mathrm{x}^2 - 116\mathrm{x} + \mathrm{x} + 3364\)
• Combine like terms: \(\mathrm{k} = \mathrm{x}^2 - 115\mathrm{x} + 3364\)
• Standard form: \(0 = \mathrm{x}^2 - 115\mathrm{x} + (3364 - \mathrm{k})\)

5. INFER the condition for exactly one solution

• A quadratic has exactly one real solution when discriminant = 0
• For \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\): discriminant = \(\mathrm{b}^2 - 4\mathrm{ac}\)
• Here: \(\mathrm{a} = 1\), \(\mathrm{b} = -115\), \(\mathrm{c} = 3364 - \mathrm{k}\)
• Set discriminant to zero: \((-115)^2 - 4(1)(3364 - \mathrm{k}) = 0\)

6. SIMPLIFY the discriminant equation

• \(13,225 - 4(3364 - \mathrm{k}) = 0\) (use calculator for \(115^2\))
• \(13,225 - 13,456 + 4\mathrm{k} = 0\)
• \(4\mathrm{k} - 231 = 0\)
• \(4\mathrm{k} = 231\)

7. APPLY CONSTRAINTS to verify the solution

• When \(\mathrm{k} = 57.75\), the solution is \(\mathrm{x} = 57.5\)
• Check domain: \(58 - 57.5 = 0.5 \gt 0\) ✓
• This confirms our solution is not extraneous

Answer: 231

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "exactly one real solution" means setting the discriminant equal to zero. Instead, they might try to solve the original equation directly or get confused about how to use the constraint. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding \((58 - \mathrm{x})^2\), particularly with the middle term \(-2(58)\mathrm{x} = -116\mathrm{x}\), or when combining like terms to get \(\mathrm{x}^2 - 115\mathrm{x} + 3364\). A common mistake is getting \(\mathrm{x}^2 - 117\mathrm{x} + 3364\) instead of the correct \(\mathrm{x}^2 - 115\mathrm{x} + 3364\). This leads them to set up the wrong discriminant equation and get an incorrect final answer.

The Bottom Line:

This problem combines multiple challenging concepts: radical equations, quadratic discriminants, and constraint application. Success requires recognizing the strategic connection between "exactly one solution" and discriminant conditions, then executing complex algebraic manipulations accurately.