\((\mathrm{x} + 4)^2 = \mathrm{x}^2 + 2\mathrm{x} + 16\)How many solutions does the given equation have?

1. INFER the approach

• The equation looks quadratic on the left but has mixed terms on the right
• Strategy: Expand the perfect square, then simplify to see what type of equation we actually have
• This will reveal the true number of solutions

2. SIMPLIFY by expanding the perfect square

• Given: \(\mathrm{(x + 4)^2 = x^2 + 2x + 16}\)
• Apply \(\mathrm{(a + b)^2 = a^2 + 2ab + b^2}\):
  \(\mathrm{(x + 4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16}\)
• Equation becomes: \(\mathrm{x^2 + 8x + 16 = x^2 + 2x + 16}\)

3. SIMPLIFY by eliminating like terms

• Subtract \(\mathrm{x^2}\) from both sides: \(\mathrm{8x + 16 = 2x + 16}\)
• Subtract 16 from both sides: \(\mathrm{8x = 2x}\)
• Subtract \(\mathrm{2x}\) from both sides: \(\mathrm{6x = 0}\)
• Divide by 6: \(\mathrm{x = 0}\)

4. INFER the number of solutions

• We get exactly one value: \(\mathrm{x = 0}\)
• This means the equation has exactly one solution

Answer: B - Exactly one

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly expand \(\mathrm{(x + 4)^2}\)

Many students remember the perfect square formula incompletely and write:
\(\mathrm{(x + 4)^2 = x^2 + 4x + 16}\) (missing the factor of 2 in the middle term)

This leads to: \(\mathrm{x^2 + 4x + 16 = x^2 + 2x + 16}\)
Simplifying: \(\mathrm{4x = 2x}\), so \(\mathrm{2x = 0}\), giving \(\mathrm{x = 0}\)

While they still get \(\mathrm{x = 0}\), they might doubt their work or make additional errors, potentially leading to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize this simplifies to a linear equation

Some students see the \(\mathrm{x^2}\) terms on both sides and assume this must be a quadratic equation with either zero or two solutions. They might try to force a quadratic formula approach or assume there should be two solutions because it "looks quadratic."

This leads them to select Choice C (Exactly two) based on the misconception that equations with \(\mathrm{x^2}\) terms always have two solutions.

The Bottom Line:

This problem tests whether students can see through the apparent complexity of a quadratic-looking equation to recognize that systematic simplification reveals a simple linear equation with exactly one solution.