z^2 - 5z - 24 = 0 What is the sum of the solutions to the given equation?...

1. TRANSLATE the problem information

• Given: \(\mathrm{z^2 - 5z - 24 = 0}\)
• Find: The sum of the solutions to this equation

2. INFER the best approach

• I can factor this quadratic and find individual solutions, then add them
• I need two numbers that multiply to -24 and add to -5
• Let me systematically check factor pairs of 24: (1,24), (2,12), (3,8), (4,6)

3. SIMPLIFY to find the correct factor pair

• Try 8 and -3: \(\mathrm{8 × (-3) = -24}\) ✓ and \(\mathrm{8 + (-3) = 5 ≠ -5}\) ✗
• Try -8 and 3: \(\mathrm{(-8) × 3 = -24}\) ✓ and \(\mathrm{(-8) + 3 = -5}\) ✓

So the factored form is: \(\mathrm{(z - 8)(z + 3) = 0}\)

4. INFER using the zero product property

• If \(\mathrm{(z - 8)(z + 3) = 0}\), then either \(\mathrm{z - 8 = 0}\) or \(\mathrm{z + 3 = 0}\)

5. SIMPLIFY to find each solution

• From \(\mathrm{z - 8 = 0}\): \(\mathrm{z = 8}\)
• From \(\mathrm{z + 3 = 0}\): \(\mathrm{z = -3}\)

6. SIMPLIFY to find the sum

• Sum = \(\mathrm{8 + (-3) = 5}\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when finding the factor pair or solving the linear equations.

For example, they might correctly identify that they need numbers multiplying to -24 and adding to -5, but incorrectly conclude that 8 and -3 work because \(\mathrm{8 + (-3) = 5}\) (forgetting they need -5). This leads them to write \(\mathrm{(z - 8)(z - 3) = 0}\), giving solutions \(\mathrm{z = 8}\) and \(\mathrm{z = 3}\), with \(\mathrm{sum = 11}\). This causes confusion since 11 typically isn't among answer choices, leading to guessing.

Second Most Common Error:

Poor INFER strategy: Students don't recognize factoring as an efficient approach and attempt to use the quadratic formula unnecessarily.

While this can work, it increases chances for arithmetic errors with the formula \(\mathrm{z = \frac{5 ± \sqrt{25 + 96}}{2} = \frac{5 ± \sqrt{121}}{2} = \frac{5 ± 11}{2}}\), giving \(\mathrm{z = 8}\) or \(\mathrm{z = -3}\). The extra complexity often leads to computational mistakes and wrong final answers.

The Bottom Line:

This problem tests whether students can systematically find factor pairs and pay careful attention to signs. The key insight is recognizing that one factor must be positive and one negative to get a negative product (-24) and negative sum (-5).