If (t^2 - 25)/(t - 5) = 9 and t neq 5, what is the value of t?

1. TRANSLATE the problem information

• Given equation: \(\frac{\mathrm{t}^2 - 25}{\mathrm{t} - 5} = 9\)
• Domain restriction: \(\mathrm{t} \neq 5\)
• Find: value of t

2. INFER the solution strategy

• The numerator \(\mathrm{t}^2 - 25\) looks like a difference of squares pattern
• If we can factor it, we might be able to cancel with the denominator \(\mathrm{t} - 5\)
• This would simplify the rational equation to something much easier to solve

3. SIMPLIFY by factoring the numerator

• Recognize \(\mathrm{t}^2 - 25\) as a difference of squares: \(\mathrm{t}^2 - 5^2\)
• Factor: \(\mathrm{t}^2 - 25 = (\mathrm{t} - 5)(\mathrm{t} + 5)\)
• Rewrite equation: \(\frac{(\mathrm{t} - 5)(\mathrm{t} + 5)}{\mathrm{t} - 5} = 9\)

4. SIMPLIFY by cancelling common factors

• Since \(\mathrm{t} \neq 5\), we can safely cancel \(\mathrm{t} - 5\) from numerator and denominator
• Result: \(\mathrm{t} + 5 = 9\)

5. SIMPLIFY the linear equation

• Solve: \(\mathrm{t} + 5 = 9\)
• Therefore: \(\mathrm{t} = 9 - 5 = 4\)

6. APPLY CONSTRAINTS to verify the solution

• Check: Does \(\mathrm{t} = 4\) satisfy \(\mathrm{t} \neq 5\)? Yes ✓
• Our solution is valid

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Difference of squares pattern

Students may not recognize that \(\mathrm{t}^2 - 25\) can be factored as \((\mathrm{t} - 5)(\mathrm{t} + 5)\). Without this insight, they might attempt to solve the rational equation directly by cross-multiplying or other complex methods, leading to unnecessary complications and potential arithmetic errors. This confusion often leads to abandoning systematic solution and guessing.

Second Most Common Error:

Weak INFER skill: Not recognizing the cancellation opportunity

Even if students factor correctly, they might not realize that the \((\mathrm{t} - 5)\) terms can cancel. Instead, they might expand everything and try to solve a more complex equation, or they might worry about "losing" the \((\mathrm{t} - 5)\) factor. This leads to overcomplication and potential selection of Choice (D) 9 by treating the equation as \(\mathrm{t}^2 - 25 = 9(\mathrm{t} - 5)\).

The Bottom Line:

This problem rewards pattern recognition and strategic thinking. Students who see the difference of squares immediately and understand safe cancellation of rational expressions will solve this in seconds, while those who don't recognize these patterns will struggle with unnecessary algebra.