What is the positive solution to the given equation?5x^2 - 37x - 24 = 0

1. TRANSLATE the problem information

• Given equation: \(5\mathrm{x}^2 - 37\mathrm{x} - 24 = 0\)
• Need to find: the positive solution

2. INFER the solution strategy

• This is a quadratic equation that needs to be solved
• Since the coefficients aren't immediately obvious for simple factoring, I'll try factoring by grouping
• I need to find two numbers that multiply to \(\mathrm{ac} = (5)(-24) = -120\) and add to \(\mathrm{b} = -37\)

3. SIMPLIFY by finding the factoring numbers

• Looking for two numbers: product = -120, sum = -37
• The numbers -40 and 3 work: \((-40)(3) = -120\) and \((-40) + 3 = -37\)

4. SIMPLIFY by rewriting and factoring

• Rewrite the middle term: \(5\mathrm{x}^2 - 40\mathrm{x} + 3\mathrm{x} - 24 = 0\)
• Group terms: \(5\mathrm{x}(\mathrm{x} - 8) + 3(\mathrm{x} - 8) = 0\)
• Factor out common factor: \((5\mathrm{x} + 3)(\mathrm{x} - 8) = 0\)

5. INFER how to use zero product property

• If \((5\mathrm{x} + 3)(\mathrm{x} - 8) = 0\), then either factor equals zero
• This gives us: \(5\mathrm{x} + 3 = 0\) OR \(\mathrm{x} - 8 = 0\)

6. SIMPLIFY each linear equation

• From \(5\mathrm{x} + 3 = 0\): subtract 3 from both sides → \(5\mathrm{x} = -3\) → \(\mathrm{x} = -\frac{3}{5}\)
• From \(\mathrm{x} - 8 = 0\): add 8 to both sides → \(\mathrm{x} = 8\)

7. APPLY CONSTRAINTS to select the final answer

• The two solutions are \(\mathrm{x} = -\frac{3}{5}\) and \(\mathrm{x} = 8\)
• Since the problem asks for the positive solution: \(\mathrm{x} = 8\)

Answer: C. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle with factoring by grouping or make arithmetic errors when finding the two numbers that multiply to -120 and add to -37.

They might try incorrect factor pairs or make sign errors during the grouping process. This leads to either an incorrect factorization or getting stuck completely, causing them to guess randomly among the answer choices.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students successfully find both solutions (\(\mathrm{x} = -\frac{3}{5}\) and \(\mathrm{x} = 8\)) but fail to recognize that the problem specifically asks for the "positive solution."

They might select the first solution they calculated or get confused about which solution to choose. This may lead them to select Choice A (3/5) if they confuse \(-\frac{3}{5}\) with \(\frac{3}{5}\), or they might randomly guess between their two solutions.

The Bottom Line:

This problem tests both algebraic manipulation skills (factoring) and careful reading comprehension (identifying the positive solution). Success requires methodical factoring technique combined with attention to the specific constraint requested.