x^2 + 6x + 4 Which of the following is equivalent to the expression above?...

1. INFER the approach needed

• Given: \(\mathrm{x^2 + 6x + 4}\) in standard form
• Answer choices: All in vertex form \(\mathrm{(x ± h)^2 ± k}\)
• Strategy: Complete the square to convert standard → vertex form

2. INFER what value completes the square

• Look at the coefficient of x: it's 6
• Take half: \(\mathrm{6 ÷ 2 = 3}\)
• Square it: \(\mathrm{3^2 = 9}\)
• This means we need to add and subtract 9

3. SIMPLIFY by adding and subtracting the completion term

• Original: \(\mathrm{x^2 + 6x + 4}\)
• Add and subtract 9: \(\mathrm{x^2 + 6x + 9 - 9 + 4}\)
• Rearrange: \(\mathrm{x^2 + 6x + 9 + 4 - 9}\)

4. SIMPLIFY by forming the perfect square

• The first three terms form a perfect square: \(\mathrm{(x + 3)^2}\)
• So we have: \(\mathrm{(x + 3)^2 + 4 - 9}\)

5. SIMPLIFY the final constants

• Combine: \(\mathrm{4 - 9 = -5}\)
• Final result: \(\mathrm{(x + 3)^2 - 5}\)

Answer: B. \(\mathrm{(x + 3)^2 - 5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to complete the square, or they try to expand the answer choices instead of transforming the given expression.

When students expand \(\mathrm{(x + 3)^2 - 5}\), they get \(\mathrm{x^2 + 6x + 9 - 5 = x^2 + 6x + 4}\), confirming it matches. But expanding all four choices takes much longer and increases error chances.

This backward approach may work but leads to Choice A if they make a sign error during expansion.

Second Most Common Error:

Poor SIMPLIFY execution: Students know to complete the square but make arithmetic mistakes in the process.

Common errors include:

• Using wrong completion term (like adding 36 instead of 9)
• Sign errors when forming the perfect square trinomial
• Arithmetic mistakes when combining final constants (-9 + 4 = -13 instead of -5)

These calculation errors typically lead them to select Choice A (\(\mathrm{(x + 3)^2 + 5}\)) or cause confusion leading to guessing.

The Bottom Line:

This problem tests whether students can fluently execute completing the square - a multi-step algebraic process requiring both strategic thinking and careful arithmetic. The vertex form answer choices provide a clear signal about the approach needed, but students must execute the technique accurately to avoid the attractive wrong answers.