6x^2 + 11x = 35 What is the positive solution to the given equation?...

1. TRANSLATE the equation to standard form

• Given: \(6\mathrm{x}^2 + 11\mathrm{x} = 35\)
• Rearrange by subtracting 35 from both sides: \(6\mathrm{x}^2 + 11\mathrm{x} - 35 = 0\)
• Now we have the standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\) where \(\mathrm{a} = 6\), \(\mathrm{b} = 11\), \(\mathrm{c} = -35\)

2. INFER the solution strategy

• Since this quadratic doesn't factor easily, we'll use the quadratic formula
• The quadratic formula will give us both positive and negative solutions

3. SIMPLIFY using the quadratic formula

• Apply \(\mathrm{x} = \frac{-\mathrm{b} \pm \sqrt{\mathrm{b}^2 - 4\mathrm{ac}}}{2\mathrm{a}}\)
• Substitute our values: \(\mathrm{x} = \frac{-11 \pm \sqrt{11^2 - 4(6)(-35)}}{2(6)}\)
• Calculate the discriminant: \(\mathrm{x} = \frac{-11 \pm \sqrt{121 + 840}}{12}\)
• SIMPLIFY further:
  \(\mathrm{x} = \frac{-11 \pm \sqrt{961}}{12}\)
  \(\mathrm{x} = \frac{-11 \pm 31}{12}\)

4. SIMPLIFY to find both solutions

• First solution:
  \(\mathrm{x} = \frac{-11 + 31}{12}\)
  \(\mathrm{x} = \frac{20}{12}\)
  \(\mathrm{x} = \frac{5}{3}\)
• Second solution:
  \(\mathrm{x} = \frac{-11 - 31}{12}\)
  \(\mathrm{x} = \frac{-42}{12}\)
  \(\mathrm{x} = -\frac{7}{2}\)

5. APPLY CONSTRAINTS to select the final answer

• The problem asks specifically for the positive solution
• Compare our solutions: \(\frac{5}{3}\) is positive, \(-\frac{7}{2}\) is negative
• Therefore, the positive solution is \(\frac{5}{3}\)

Answer: B (5/3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when calculating the discriminant or when simplifying the final fractions.

The discriminant calculation involves \(11^2 - 4(6)(-35) = 121 + 840 = 961\), and students might calculate this as 121 - 840 or make other sign errors. Similarly, when simplifying 20/12, students might not reduce it properly to 5/3.

This leads to selecting an incorrect answer choice or getting confused and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students find both solutions correctly but fail to identify which one is positive.

Students might calculate both \(\mathrm{x} = \frac{5}{3}\) and \(\mathrm{x} = -\frac{7}{2}\) but then select the wrong solution, not carefully reading that the problem asks for the positive solution specifically.

This may lead them to select a different answer choice or abandon the problem entirely.

The Bottom Line:

This problem tests whether students can systematically apply the quadratic formula while maintaining accuracy through multiple algebraic steps and then apply the constraint of selecting only the positive solution.