\((2\mathrm{x} + 1)^2 = (2\mathrm{x} - 1)^2 + 16\)Which of the following is a solution to the given equation?-{2}023

1. INFER the solution strategy

• Given: \((2\mathrm{x} + 1)^2 = (2\mathrm{x} - 1)^2 + 16\)
• Strategy: Since both sides contain perfect squares, expand them to eliminate the squared terms and create a linear equation

2. SIMPLIFY by expanding the left side

• Use the formula \((\mathrm{a} + \mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\)
• \((2\mathrm{x} + 1)^2 = (2\mathrm{x})^2 + 2(2\mathrm{x})(1) + 1^2\)
• \(= 4\mathrm{x}^2 + 4\mathrm{x} + 1\)

3. SIMPLIFY by expanding the right side

• Use the formula \((\mathrm{a} - \mathrm{b})^2 = \mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2\)
• \((2\mathrm{x} - 1)^2 + 16 = (2\mathrm{x})^2 - 2(2\mathrm{x})(1) + 1^2 + 16\)
• \(= 4\mathrm{x}^2 - 4\mathrm{x} + 1 + 16\)
• \(= 4\mathrm{x}^2 - 4\mathrm{x} + 17\)

4. SIMPLIFY by setting equal and solving

• \(4\mathrm{x}^2 + 4\mathrm{x} + 1 = 4\mathrm{x}^2 - 4\mathrm{x} + 17\)
• Subtract \(4\mathrm{x}^2\) from both sides: \(4\mathrm{x} + 1 = -4\mathrm{x} + 17\)
• Add \(4\mathrm{x}\) to both sides: \(8\mathrm{x} + 1 = 17\)
• Subtract 1 from both sides: \(8\mathrm{x} = 16\)
• Divide by 8: \(\mathrm{x} = 2\)

Answer: C. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when expanding \((2\mathrm{x} - 1)^2\)

Many students correctly expand \((2\mathrm{x} + 1)^2 = 4\mathrm{x}^2 + 4\mathrm{x} + 1\), but when expanding \((2\mathrm{x} - 1)^2\), they forget that the middle term should be negative. They might write \((2\mathrm{x} - 1)^2 = 4\mathrm{x}^2 + 4\mathrm{x} + 1\) instead of \(4\mathrm{x}^2 - 4\mathrm{x} + 1\). This leads to the equation \(4\mathrm{x}^2 + 4\mathrm{x} + 1 = 4\mathrm{x}^2 + 4\mathrm{x} + 1 + 16\), which simplifies to \(0 = 16\) (impossible), causing confusion and potentially leading them to select Choice B (0) thinking maybe they made an error and the solution is the "neutral" value.

Second Most Common Error:

Poor INFER reasoning: Students try to solve by taking square roots of both sides immediately

Some students see the squared terms and think they should take the square root of both sides right away: \(\sqrt{(2\mathrm{x} + 1)^2} = \sqrt{(2\mathrm{x} - 1)^2 + 16}\). This doesn't work because the right side isn't a perfect square, and they can't simplify \(\sqrt{(2\mathrm{x} - 1)^2 + 16}\). This leads to confusion and guessing, or they might incorrectly assume the equation has no solution and guess randomly.

The Bottom Line:

This problem tests whether students can systematically expand perfect squares and maintain accuracy with signs. The key insight is recognizing that expansion eliminates the squared terms, turning a seemingly complex equation into a straightforward linear equation.