What is the positive solution to the given equation? -{4x^2 - 7x = -36}...

1. TRANSLATE the problem information

• Given equation: \(-4\mathrm{x}^2 - 7\mathrm{x} = -36\)
• Need to find: the positive solution (not both solutions)

2. INFER the solution approach

• This is a quadratic equation, so we expect two solutions
• We need to use a systematic method: quadratic formula, factoring, or completing the square
• The quadratic formula works for any quadratic equation

3. SIMPLIFY to standard form

• Move all terms to one side: \(-4\mathrm{x}^2 - 7\mathrm{x} + 36 = 0\)
• Multiply by -1 to make leading coefficient positive: \(4\mathrm{x}^2 + 7\mathrm{x} - 36 = 0\)

4. SIMPLIFY using the quadratic formula

• For \(4\mathrm{x}^2 + 7\mathrm{x} - 36 = 0\), we have \(\mathrm{a} = 4, \mathrm{b} = 7, \mathrm{c} = -36\)
• \(\mathrm{x} = \frac{-7 \pm \sqrt{7^2 - 4(4)(-36)}}{2(4)}\)
• \(\mathrm{x} = \frac{-7 \pm \sqrt{49 + 576}}{8}\)
• \(\mathrm{x} = \frac{-7 \pm \sqrt{625}}{8}\)
• \(\mathrm{x} = \frac{-7 \pm 25}{8}\)

5. SIMPLIFY to find both solutions

• \(\mathrm{x} = \frac{-7 + 25}{8} = \frac{18}{8} = \frac{9}{4}\)
• \(\mathrm{x} = \frac{-7 - 25}{8} = \frac{-32}{8} = -4\)

6. APPLY CONSTRAINTS to select final answer

• Since the question asks for the positive solution: \(\mathrm{x} = \frac{9}{4}\)

Answer: B. 9/4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors when applying the quadratic formula, especially when computing the discriminant or simplifying fractions.

Students might calculate \(\sqrt{49 + 576}\) incorrectly, or make sign errors when computing \(\frac{-7 \pm 25}{8}\). These calculation mistakes lead to wrong values that don't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Finding both solutions correctly but failing to identify which one the question is asking for.

Students solve the quadratic equation and get both 9/4 and -4, but don't carefully read that the question asks for the "positive solution." They might select the first solution they calculated or get confused about which to choose. This may lead them to select Choice A (7/4) if they made a computational error, or get stuck between their two correct solutions.

The Bottom Line:

This problem tests careful algebraic execution combined with reading comprehension. Students must not only solve a quadratic equation accurately but also pay attention to the constraint that specifies which solution to report.