x^2 - 10x + 13 = 0 Which of the following is a solution to the given equation?...

1. INFER the solution approach

• This is a quadratic equation in standard form \(\mathrm{ax^2 + bx + c = 0}\)
• Since it doesn't factor easily, the quadratic formula is the most reliable method
• We need: \(\mathrm{a = 1, b = -10, c = 13}\)

2. SIMPLIFY using the quadratic formula

• Apply \(\mathrm{x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}}\)
• Substitute: \(\mathrm{x = \frac{-(-10) ± \sqrt{(-10)^2 - 4(1)(13)}}{2(1)}}\)
• SIMPLIFY the calculation: \(\mathrm{x = \frac{10 ± \sqrt{100 - 52}}{2}}\)
• This gives us: \(\mathrm{x = \frac{10 ± \sqrt{48}}{2}}\)

3. SIMPLIFY the radical expression

• We need to simplify \(\mathrm{\sqrt{48}}\) by finding perfect square factors
• \(\mathrm{48 = 16 × 3 = 4^2 × 3}\)
• Therefore: \(\mathrm{\sqrt{48} = \sqrt{4^2 × 3} = 4\sqrt{3}}\)

4. SIMPLIFY to final form

• Substitute back: \(\mathrm{x = \frac{10 ± 4\sqrt{3}}{2}}\)
• Factor out the common factor: \(\mathrm{x = 5 ± 2\sqrt{3}}\)
• The two solutions are: \(\mathrm{x = 5 + 2\sqrt{3}}\) and \(\mathrm{x = 5 - 2\sqrt{3}}\)

5. INFER which answer choice matches

• Compare with given options
• Choice D: \(\mathrm{5 - 2\sqrt{3}}\) matches our second solution

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make errors when simplifying \(\mathrm{\sqrt{48}}\), either not recognizing the perfect square factor 16, or making arithmetic mistakes in the radical simplification.

For example, they might leave \(\mathrm{\sqrt{48}}\) unsimplified, getting \(\mathrm{x = \frac{10 ± \sqrt{48}}{2}}\), which doesn't match any answer choice. Or they might incorrectly simplify \(\mathrm{\sqrt{48}}\) as \(\mathrm{2\sqrt{12}}\) or \(\mathrm{6\sqrt{2}}\), leading to wrong final expressions that cause confusion and guessing.

Second Most Common Error:

Arithmetic errors in quadratic formula: Students may make sign errors or calculation mistakes when computing \(\mathrm{b^2 - 4ac = 100 - 52 = 48}\), or when applying the formula itself.

These computational errors can lead to completely different radical expressions under the square root, causing them to select an incorrect choice or abandon the systematic solution and guess.

The Bottom Line:

This problem tests both formula application and radical manipulation skills. Success requires careful attention to arithmetic details, especially in radical simplification where recognizing perfect square factors is crucial for matching the answer format.