What is a solution to the given equation?\(\mathrm{(d - 30)(d + 30) - 7 = -7}\)

1. SIMPLIFY the equation to standard form

• Given: \((\mathrm{d} - 30)(\mathrm{d} + 30) - 7 = -7\)
• Add 7 to both sides: \((\mathrm{d} - 30)(\mathrm{d} + 30) = 0\)
• Now we have a product that equals zero

2. INFER the solution strategy

• When a product of two factors equals zero, at least one factor must be zero
• This means either \((\mathrm{d} - 30) = 0\) OR \((\mathrm{d} + 30) = 0\)
• We need to solve both possibilities to find all solutions

3. SIMPLIFY each linear equation

• For \(\mathrm{d} - 30 = 0\): Add 30 to both sides → \(\mathrm{d} = 30\)
• For \(\mathrm{d} + 30 = 0\): Subtract 30 from both sides → \(\mathrm{d} = -30\)

4. CONSIDER ALL CASES to provide complete answer

• Both solutions are valid: \(\mathrm{d} = 30\) and \(\mathrm{d} = -30\)

Answer: 30, -30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students often make errors in the initial step of adding 7 to both sides, either forgetting to add it to both sides or making arithmetic mistakes.

Instead of getting \((\mathrm{d} - 30)(\mathrm{d} + 30) = 0\), they might incorrectly have \((\mathrm{d} - 30)(\mathrm{d} + 30) = -14\) or some other value. This prevents them from using the zero product property and leads to confusion and guessing.

Second Most Common Error:

Missing INFER reasoning: Students may successfully simplify to \((\mathrm{d} - 30)(\mathrm{d} + 30) = 0\) but fail to recognize that they should use the zero product property.

They might attempt to expand the left side or try other unnecessary algebraic manipulations instead of setting each factor equal to zero. This leads them to get stuck and abandon systematic solution.

The Bottom Line:

This problem tests whether students can recognize when to apply the zero product property after performing basic algebraic simplification. The key insight is that getting a product equal to zero is a signal to set each factor to zero rather than continuing with other algebraic operations.