Question:In a right triangle, the legs have lengths x and x + 1, where x is a positive real number....

1. TRANSLATE the problem information

• Given information:
  • Right triangle with legs of length \(\mathrm{x}\) and \(\mathrm{x + 1}\)
  • Hypotenuse has length \(\mathrm{x + 3}\)
  • \(\mathrm{x}\) is a positive real number
• We need to find the value of \(\mathrm{x}\)

2. INFER the mathematical approach

• Since we have a right triangle with known side lengths, we can use the Pythagorean theorem
• Set up the equation: \(\mathrm{(leg_1)^2 + (leg_2)^2 = (hypotenuse)^2}\)
• This gives us: \(\mathrm{x^2 + (x + 1)^2 = (x + 3)^2}\)

3. SIMPLIFY by expanding and combining terms

• Expand each squared term:
  • \(\mathrm{x^2}\) stays as \(\mathrm{x^2}\)
  • \(\mathrm{(x + 1)^2 = x^2 + 2x + 1}\)
  • \(\mathrm{(x + 3)^2 = x^2 + 6x + 9}\)
• Our equation becomes: \(\mathrm{x^2 + x^2 + 2x + 1 = x^2 + 6x + 9}\)
• Combine like terms: \(\mathrm{2x^2 + 2x + 1 = x^2 + 6x + 9}\)
• Move everything to one side: \(\mathrm{x^2 - 4x - 8 = 0}\)

4. SIMPLIFY using the quadratic formula

• For \(\mathrm{x^2 - 4x - 8 = 0}\), we have \(\mathrm{a = 1}\), \(\mathrm{b = -4}\), \(\mathrm{c = -8}\)
• Apply the quadratic formula: \(\mathrm{x = \frac{4 ± \sqrt{16 + 32}}{2}}\)
• SIMPLIFY the radical: \(\mathrm{x = \frac{4 ± \sqrt{48}}{2} = \frac{4 ± 4\sqrt{3}}{2} = 2 ± 2\sqrt{3}}\)

5. APPLY CONSTRAINTS to select the final answer

• Since \(\mathrm{x}\) represents a side length, it must be positive
• We have two solutions: \(\mathrm{x = 2 + 2\sqrt{3}}\) and \(\mathrm{x = 2 - 2\sqrt{3}}\)
• Since \(\mathrm{2\sqrt{3} ≈ 3.46}\), we have \(\mathrm{x = 2 - 2\sqrt{3} ≈ -1.46}\) (negative)
• Therefore: \(\mathrm{x = 2 + 2\sqrt{3}}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make algebraic errors when expanding \(\mathrm{(x + 1)^2}\) and \(\mathrm{(x + 3)^2}\), often forgetting the middle terms or making sign errors when moving terms to one side of the equation.

For example, they might incorrectly expand \(\mathrm{(x + 1)^2}\) as \(\mathrm{x^2 + 1}\) instead of \(\mathrm{x^2 + 2x + 1}\), or make errors when rearranging the equation. These computational mistakes lead to an incorrect quadratic equation and ultimately to selecting the wrong answer choice or getting confused and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic equation to get \(\mathrm{x = 2 ± 2\sqrt{3}}\), but fail to recognize that \(\mathrm{x}\) must be positive since it represents a side length.

They might select \(\mathrm{x = 2 - 2\sqrt{3}}\) without checking if this value makes geometric sense, or they might get confused about which root to choose. Since \(\mathrm{2 - 2\sqrt{3}}\) is negative and none of the answer choices match this value, this leads to confusion and random guessing among the given options.

The Bottom Line:

This problem tests both algebraic manipulation skills and geometric reasoning. Success requires careful expansion of algebraic expressions, systematic equation solving, and thoughtful application of real-world constraints to mathematical solutions.