sqrt(x + 4) = 11 What value of x satisfies the equation above?...

1. TRANSLATE the problem information

• Given equation: \(\sqrt{\mathrm{x + 4}} = 11\)
• Need to find: value of x

2. INFER the solution strategy

• To solve an equation with a square root, we need to eliminate the square root
• The most direct approach is to square both sides of the equation
• This works because \((\sqrt{\mathrm{a}})^2 = \mathrm{a}\)

3. SIMPLIFY by squaring both sides

• Square the left side: \((\sqrt{\mathrm{x + 4}})^2 = \mathrm{x + 4}\)
• Square the right side: \(11^2 = 121\)
• New equation: \(\mathrm{x + 4 = 121}\)

4. SIMPLIFY to isolate x

• Subtract 4 from both sides: \(\mathrm{x + 4 - 4 = 121 - 4}\)
• Final result: \(\mathrm{x = 117}\)

Answer: 117

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that both sides need to be squared, or forgetting to square the right side completely.

Students might square only the left side and write: \(\mathrm{x + 4 = 11}\), leading to \(\mathrm{x = 7}\). When they check this answer (\(\sqrt{7 + 4} = \sqrt{11} \approx 3.3\)), it doesn't equal 11, causing confusion and potential guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic mistakes when calculating \(11^2 = 121\).

Some students might incorrectly calculate \(11^2\) as 111 or make other computational errors, leading to wrong final answers. They might also add 4 instead of subtracting: \(\mathrm{x = 121 + 4 = 125}\).

The Bottom Line:

The key insight is recognizing that squaring both sides simultaneously eliminates the square root while preserving the equality. Students who miss this strategic step or make basic arithmetic errors will struggle to reach the correct answer.