sqrt(k) + 8 = 20 What is the solution to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(\sqrt{\mathrm{k}} + 8 = 20\)
• Need to find: The value of k

2. INFER the solution strategy

• To solve for k inside a square root, we need to isolate the radical term first
• Then we can square both sides to eliminate the square root

3. SIMPLIFY by isolating the square root

• Start with: \(\sqrt{\mathrm{k}} + 8 = 20\)
• Subtract 8 from both sides: \(\sqrt{\mathrm{k}} = 20 - 8\)
• This gives us: \(\sqrt{\mathrm{k}} = 12\)

4. SIMPLIFY by squaring both sides

• Square both sides: \((\sqrt{\mathrm{k}})^2 = 12^2\)
• This eliminates the square root: \(\mathrm{k} = 144\)

Answer: 144 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students stop after finding \(\sqrt{\mathrm{k}} = 12\) and think \(\mathrm{k} = 12\)

They forget that k is the value under the square root, not the square root itself. Since \(\sqrt{\mathrm{k}} = 12\), we need \(\mathrm{k} = 12^2 = 144\).

This may lead them to select Choice A (12)

Second Most Common Error:

Poor INFER reasoning about order of operations: Students try to square the entire left side as \((\sqrt{\mathrm{k}} + 8)^2 = 20^2\)

This creates the much more complex equation \(\mathrm{k} + 16\sqrt{\mathrm{k}} + 64 = 400\), leading to confusion and wrong calculations.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key insight is recognizing that when you have \(\sqrt{\mathrm{k}} = \text{some number}\), you must square both sides to find k. Many students confuse the square root of k with k itself.