What is the positive solution to the given equation?7x^2 - 20x - 32 = 0

1. INFER the solution approach

• Given equation: \(7\mathrm{x}^2 - 20\mathrm{x} - 32 = 0\)
• This is a quadratic equation that needs to be solved for x
• Since the coefficient of x² is not 1, factoring by grouping will be most efficient

2. INFER the factoring strategy

• For factoring by grouping, I need two numbers that:
  • Multiply to give \(\mathrm{ac} = (7)(-32) = -224\)
  • Add to give \(\mathrm{b} = -20\)
• Testing factors: -28 and 8 work because \((-28)(8) = -224\) and \(-28 + 8 = -20\)

3. SIMPLIFY by rewriting and factoring

• Rewrite the middle term: \(7\mathrm{x}^2 - 28\mathrm{x} + 8\mathrm{x} - 32 = 0\)
• Factor by grouping: \(7\mathrm{x}(\mathrm{x} - 4) + 8(\mathrm{x} - 4) = 0\)
• Factor out common binomial: \((7\mathrm{x} + 8)(\mathrm{x} - 4) = 0\)

4. INFER using zero product property

• If \((7\mathrm{x} + 8)(\mathrm{x} - 4) = 0\), then either factor equals zero
• This gives us: \(7\mathrm{x} + 8 = 0\) OR \(\mathrm{x} - 4 = 0\)

5. SIMPLIFY to solve each equation

• From \(7\mathrm{x} + 8 = 0\): \(7\mathrm{x} = -8\), so \(\mathrm{x} = -\frac{8}{7}\)
• From \(\mathrm{x} - 4 = 0\): \(\mathrm{x} = 4\)

6. APPLY CONSTRAINTS to select final answer

• The problem asks for the positive solution
• Between \(\mathrm{x} = -\frac{8}{7}\) (negative) and \(\mathrm{x} = 4\) (positive)
• The positive solution is \(\mathrm{x} = 4\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the factoring by grouping method when the leading coefficient isn't 1, instead attempting to factor as \((\mathrm{ax} + \mathrm{b})(\mathrm{cx} + \mathrm{d})\) through guess-and-check. This approach becomes very time-consuming with a leading coefficient of 7, leading to frustration and potential calculation errors. This often causes them to get stuck and randomly guess an answer.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly find both solutions \((-\frac{8}{7}\) and \(4)\) but fail to notice the question specifically asks for the 'positive solution.' They might provide both answers or mistakenly think \(-\frac{8}{7}\) is positive due to a sign error. This leads to confusion about which answer to select.

The Bottom Line:

The key challenge is recognizing that factoring by grouping is the most efficient method when dealing with quadratics that have larger leading coefficients, and then carefully reading what the question is actually asking for.