Question:Consider the equation 3x^2 - 2x = 21. What is the positive value of x that satisfies this equation?

1. TRANSLATE the problem information

• Given equation: \(3\mathrm{x}^2 - 2\mathrm{x} = 21\)
• Need to find: the positive value of x
• This tells us we'll get multiple solutions but only want the positive one

2. TRANSLATE to standard form

• Move all terms to one side: \(3\mathrm{x}^2 - 2\mathrm{x} - 21 = 0\)
• Now we have the standard form \(\mathrm{a}\mathrm{x}^2 + \mathrm{b}\mathrm{x} + \mathrm{c} = 0\) where:
  • \(\mathrm{a} = 3\)
  • \(\mathrm{b} = -2\)
  • \(\mathrm{c} = -21\)

3. INFER the solution approach

• Since this quadratic doesn't factor easily, we need the quadratic formula
• The quadratic formula will give us two solutions due to the ± symbol
• We'll need to identify which solution is positive

4. SIMPLIFY using the quadratic formula

• \(\mathrm{x} = \frac{-\mathrm{b} ± \sqrt{\mathrm{b}^2 - 4\mathrm{a}\mathrm{c}}}{2\mathrm{a}}\)
• \(\mathrm{x} = \frac{-(-2) ± \sqrt{(-2)^2 - 4(3)(-21)}}{2(3)}\)
• \(\mathrm{x} = \frac{2 ± \sqrt{4 + 252}}{6}\)
• \(\mathrm{x} = \frac{2 ± \sqrt{256}}{6}\)
• \(\mathrm{x} = \frac{2 ± 16}{6}\)

5. SIMPLIFY to find both solutions

• \(\mathrm{x} = \frac{2 + 16}{6} = \frac{18}{6} = 3\)
• \(\mathrm{x} = \frac{2 - 16}{6} = \frac{-14}{6} = -\frac{7}{3}\)

6. APPLY CONSTRAINTS to select the final answer

• The problem asks for the positive value of x
• Between \(\mathrm{x} = 3\) and \(\mathrm{x} = -\frac{7}{3}\), only \(\mathrm{x} = 3\) is positive

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may try to solve \(3\mathrm{x}^2 - 2\mathrm{x} = 21\) without first rearranging to standard form. They might attempt to factor or apply the quadratic formula incorrectly, leading to computational errors and incorrect solutions. This leads to confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic and find both \(\mathrm{x} = 3\) and \(\mathrm{x} = -\frac{7}{3}\), but either provide both answers or select the negative solution. They miss that the problem specifically asks for 'the positive value of x.' This may lead them to enter -7/3 or provide both solutions instead of just 3.

The Bottom Line:

This problem tests whether students can systematically work through the quadratic formula process while paying attention to the specific constraint requested in the problem statement.