Question:Let y be a real number satisfying \(2(\mathrm{y} - 5)^2 = 18\). Without solving for y, determine the value of...

1. SIMPLIFY the given constraint

• Given: \(2(\mathrm{y} - 5)^2 = 18\)
• Divide both sides by 2: \((\mathrm{y} - 5)^2 = 9\)

2. INFER the strategic approach

• We want \(\mathrm{y}^2 - 10\mathrm{y}\) but have information about \((\mathrm{y} - 5)^2\)
• Key insight: Look for a way to express \(\mathrm{y}^2 - 10\mathrm{y}\) in terms of \((\mathrm{y} - 5)^2\)
• This will let us use our constraint without solving for y

3. SIMPLIFY through algebraic rearrangement

• Expand \((\mathrm{y} - 5)^2\): \((\mathrm{y} - 5)^2 = \mathrm{y}^2 - 10\mathrm{y} + 25\)
• Rearrange: \(\mathrm{y}^2 - 10\mathrm{y} = (\mathrm{y} - 5)^2 - 25\)

4. SIMPLIFY by substitution

• Substitute \((\mathrm{y} - 5)^2 = 9\):
• \(\mathrm{y}^2 - 10\mathrm{y} = 9 - 25 = -16\)

Answer: −16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss the key strategic insight and try to solve for y explicitly first.

They see \(2(\mathrm{y} - 5)^2 = 18\), get \((\mathrm{y} - 5)^2 = 9\), then solve \(\mathrm{y} - 5 = ±3\) to find \(\mathrm{y} = 8\) or \(\mathrm{y} = 2\). Then they calculate \(\mathrm{y}^2 - 10\mathrm{y}\) for each value. While this works, it ignores the problem's hint "without solving for y" and requires more work. This approach usually gets the right answer but misses the elegant method the problem is testing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize they should work with \((\mathrm{y} - 5)^2\) but make errors in the algebraic expansion or rearrangement.

They might incorrectly expand \((\mathrm{y} - 5)^2\) or make sign errors when rearranging to isolate \(\mathrm{y}^2 - 10\mathrm{y}\). For example, getting \((\mathrm{y} - 5)^2 = \mathrm{y}^2 - 10\mathrm{y} - 25\) instead of \(\mathrm{y}^2 - 10\mathrm{y} + 25\), leading to \(\mathrm{y}^2 - 10\mathrm{y} = 9 + 25 = 34\) instead of the correct −16.

The Bottom Line:

This problem tests whether students can recognize algebraic relationships and work strategically with expressions rather than defaulting to solving equations explicitly. The key is seeing that the target expression is "hidden" within the expansion of the given constraint.