1/(x^2 + 10x + 25) = 4 If x is a solution to the given equation, which of the following...

1. INFER the structure of the denominator

• Given equation: \(\frac{1}{x^2 + 10x + 25} = 4\)
• Key insight: \(x^2 + 10x + 25\) looks like it might factor nicely
• Check if it's a perfect square: \(x^2 + 10x + 25 = (x + 5)^2\)
• This transforms our equation to: \(\frac{1}{(x + 5)^2} = 4\)

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by \((x + 5)^2\): \(1 = 4(x + 5)^2\)
• Divide both sides by 4: \(\frac{1}{4} = (x + 5)^2\)

3. SIMPLIFY by taking the square root

• Take the square root of both sides: \(\pm\frac{1}{2} = x + 5\)
• This gives us two possible values: \(x + 5 = \frac{1}{2}\) or \(x + 5 = -\frac{1}{2}\)

4. Select the answer that matches

• The question asks for "a possible value of \(x + 5\)"
• Looking at the choices, \(\frac{1}{2}\) appears as choice A

Answer: A (\(\frac{1}{2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(x^2 + 10x + 25\) is a perfect square trinomial

Students may attempt to use the quadratic formula on the original equation or get overwhelmed by the complex-looking denominator. Without seeing the \((x + 5)^2\) pattern, they might try to cross-multiply incorrectly or attempt other inefficient approaches that lead to computational errors and confusion.

This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic mistakes in the manipulation steps

Even if students recognize the perfect square, they might make errors when multiplying both sides by \((x + 5)^2\) or forget that taking a square root yields both positive and negative solutions. Calculation errors at any step can lead to incorrect values.

This may lead them to select other answer choices based on their miscalculated results.

The Bottom Line:

This problem rewards pattern recognition - seeing the perfect square trinomial immediately simplifies what appears to be a complex rational equation into a straightforward algebraic manipulation.