2x^2 - 2 = 2x + 3 Which of the following is a solution to the equation above?...

1. TRANSLATE the problem information

• Given equation: \(2\mathrm{x}^2 - 2 = 2\mathrm{x} + 3\)
• Need to find: Which value of x satisfies this equation

2. INFER the solution approach

• This is a quadratic equation that needs to be solved
• Strategy: Rearrange to standard form, then use quadratic formula
• The quadratic formula will give us the exact solutions we need

3. SIMPLIFY to standard form

• Move all terms to one side: \(2\mathrm{x}^2 - 2 - 2\mathrm{x} - 3 = 0\)
• Combine like terms: \(2\mathrm{x}^2 - 2\mathrm{x} - 5 = 0\)
• Now we have standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\) where \(\mathrm{a} = 2, \mathrm{b} = -2, \mathrm{c} = -5\)

4. SIMPLIFY using the quadratic formula

• Apply \(\mathrm{x} = \frac{-\mathrm{b} \pm \sqrt{\mathrm{b}^2 - 4\mathrm{ac}}}{2\mathrm{a}}\)
• Substitute values: \(\mathrm{x} = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(2)(-5)}}{2(2)}\)
• SIMPLIFY step by step:
  • \(\mathrm{x} = \frac{2 \pm \sqrt{4 + 40}}{4}\)
  • \(\mathrm{x} = \frac{2 \pm \sqrt{44}}{4}\)

5. SIMPLIFY the radical

• Factor out perfect squares: \(\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}\)
• Substitute back: \(\mathrm{x} = \frac{2 \pm 2\sqrt{11}}{4}\)
• Factor and reduce: \(\mathrm{x} = \frac{2(1 \pm \sqrt{11})}{4} = \frac{1 \pm \sqrt{11}}{2}\)

6. INFER which solution matches the choices

• Two solutions: \(\frac{1 + \sqrt{11}}{2}\) and \(\frac{1 - \sqrt{11}}{2}\)
• Looking at choices, \(\frac{1 + \sqrt{11}}{2}\) matches choice D exactly

Answer: D. \(\frac{1 + \sqrt{11}}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make computational errors when applying the quadratic formula, particularly when simplifying \(\sqrt{44}\). They might incorrectly write \(\sqrt{44} = 4\sqrt{11}\) instead of \(2\sqrt{11}\), or make sign errors in the formula application.

If they write \(\sqrt{44} = 4\sqrt{11}\), they get \(\mathrm{x} = \frac{2 \pm 4\sqrt{11}}{4} = \frac{1 \pm 2\sqrt{11}}{2}\), leading them to think the answer should be \(\frac{1}{2} + 2\sqrt{11}\), which looks similar to Choice C \(\frac{1}{2} + \sqrt{11}\) and might cause them to select it incorrectly.

Second Most Common Error:

Missing conceptual knowledge of quadratic formula: Some students remember an incomplete version of the quadratic formula, using \(\mathrm{x} = \frac{-\mathrm{b} \pm \sqrt{\mathrm{b}^2 - 4\mathrm{ac}}}{\mathrm{a}}\) instead of the correct \(\mathrm{x} = \frac{-\mathrm{b} \pm \sqrt{\mathrm{b}^2 - 4\mathrm{ac}}}{2\mathrm{a}}\). This missing factor of 2 in the denominator leads them to get \(\mathrm{x} = \frac{2 \pm 2\sqrt{11}}{2} = 1 \pm \sqrt{11}\), which makes Choice B \(1 - \sqrt{11}\) appear correct.

The Bottom Line:

This problem tests both conceptual knowledge of the quadratic formula and careful algebraic execution. The most challenging part is maintaining accuracy through multiple simplification steps while recognizing that only one of the two solutions will appear in the answer choices.