Question:Consider the equation 1/(x - 1) + 1/(x + 2) = 1/4.One solution to the equation can be written as...

1. TRANSLATE the equation setup

• Given equation: \(\frac{1}{\mathrm{x}-1} + \frac{1}{\mathrm{x}+2} = \frac{1}{4}\)
• Need to find: \(\mathrm{k}\) where one solution is \(\frac{7-\sqrt{\mathrm{k}}}{2}\)

2. INFER the solution strategy

• Two fractions on left side need to be combined before we can solve
• Once we have a single fraction, we can cross-multiply to eliminate denominators
• This will likely create a quadratic equation

3. SIMPLIFY by combining fractions

• Find common denominator: \((\mathrm{x}-1)(\mathrm{x}+2)\)
• Combine: \(\frac{1}{\mathrm{x}-1} + \frac{1}{\mathrm{x}+2} = \frac{(\mathrm{x}+2) + (\mathrm{x}-1)}{(\mathrm{x}-1)(\mathrm{x}+2)}\)
• SIMPLIFY the numerator: \((\mathrm{x}+2) + (\mathrm{x}-1) = 2\mathrm{x}+1\)
• SIMPLIFY the denominator: \((\mathrm{x}-1)(\mathrm{x}+2) = \mathrm{x}^2 + \mathrm{x} - 2\)

4. SIMPLIFY by cross-multiplying

• Now we have: \(\frac{2\mathrm{x}+1}{\mathrm{x}^2+\mathrm{x}-2} = \frac{1}{4}\)
• Cross-multiply: \(4(2\mathrm{x}+1) = \mathrm{x}^2 + \mathrm{x} - 2\)
• Distribute: \(8\mathrm{x} + 4 = \mathrm{x}^2 + \mathrm{x} - 2\)

5. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(\mathrm{x}^2 + \mathrm{x} - 2 - 8\mathrm{x} - 4 = 0\)
• Combine like terms: \(\mathrm{x}^2 - 7\mathrm{x} - 6 = 0\)

6. SIMPLIFY using quadratic formula

• For \(\mathrm{x}^2 - 7\mathrm{x} - 6 = 0\), we have \(\mathrm{a} = 1\), \(\mathrm{b} = -7\), \(\mathrm{c} = -6\)
• Apply formula: \(\mathrm{x} = \frac{7 \pm \sqrt{49+24}}{2} = \frac{7 \pm \sqrt{73}}{2}\) (use calculator)
• Two solutions: \(\frac{7+\sqrt{73}}{2}\) and \(\frac{7-\sqrt{73}}{2}\)

7. INFER the final answer

• Problem states one solution is \(\frac{7-\sqrt{\mathrm{k}}}{2}\)
• Comparing with \(\frac{7-\sqrt{73}}{2}\), we get \(\mathrm{k} = 73\)

Answer: 73

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when combining fractions, particularly getting the wrong common denominator or incorrectly adding the numerators.

For example, they might write \((\mathrm{x}+2) + (\mathrm{x}-1) = \mathrm{x}+1\) instead of \(2\mathrm{x}+1\), or expand \((\mathrm{x}-1)(\mathrm{x}+2)\) incorrectly. These early errors cascade through the entire solution, leading to a different quadratic equation and ultimately the wrong value of k. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about solution form: Students correctly solve the quadratic but fail to connect their answer format with the given form \(\frac{7-\sqrt{\mathrm{k}}}{2}\).

They might get \(\mathrm{x} = \frac{7 \pm \sqrt{73}}{2}\) but not recognize that \(\mathrm{k} = 73\), especially if they don't carefully read which solution form the problem is asking about. This causes them to get stuck and guess.

The Bottom Line:

This problem tests sustained algebraic accuracy through multiple steps. The rational equation setup looks straightforward, but it requires flawless execution of fraction operations, expansion, and quadratic solving to reach the correct k value.