What is one of the solutions to the given equation? z^2 + 10z - 24 = 0...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{z^2 + 10z - 24 = 0}\)
• Need to find: The solutions (values of z that make the equation true)

2. INFER the solution approach

• This is a quadratic equation in standard form
• Since the coefficient of \(\mathrm{z^2}\) is 1, factoring should work well
• Strategy: Find two numbers that multiply to -24 and add to 10

3. INFER the factor pair

• Need two numbers: \(\mathrm{(number_1)(number_2) = -24}\) and \(\mathrm{number_1 + number_2 = 10}\)
• Since the product is negative, one number must be positive and one negative
• Since the sum is positive (+10), the larger absolute value must be positive
• Testing factor pairs: \(\mathrm{12 \times (-2) = -24}\) and \(\mathrm{12 + (-2) = 10}\) ✓

4. SIMPLIFY by factoring the quadratic

• Rewrite: \(\mathrm{z^2 + 10z - 24 = (z + 12)(z - 2) = 0}\)
• Apply zero product property: if \(\mathrm{(z + 12)(z - 2) = 0}\), then \(\mathrm{z + 12 = 0}\) or \(\mathrm{z - 2 = 0}\)

5. SIMPLIFY by solving each linear equation

• From \(\mathrm{z + 12 = 0}\): \(\mathrm{z = -12}\)
• From \(\mathrm{z - 2 = 0}\): \(\mathrm{z = 2}\)

6. CONSIDER ALL CASES for the final answer

• Both solutions are valid: \(\mathrm{z = 2}\) and \(\mathrm{z = -12}\)
• Either answer would be correct for this question

Answer: 2 or -12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students struggle to find the correct factor pair that satisfies both conditions (multiply to -24 AND add to 10). They might try various combinations randomly or give up when the first few attempts don't work.

This often leads to incorrect factoring like \(\mathrm{(z + 6)(z - 4)}\) or \(\mathrm{(z + 8)(z - 3)}\), which would give wrong solutions. This leads to confusion and guessing between answer choices.

Second Most Common Error:

Incomplete CONSIDER ALL CASES execution: Students correctly factor the quadratic and find one solution but forget that quadratic equations typically have two solutions. They might stop after finding \(\mathrm{z = 2}\) and not realize \(\mathrm{z = -12}\) is also a solution.

Since the question asks for "one of the solutions," this partial approach might still get the right answer, but demonstrates incomplete understanding of quadratic solutions.

The Bottom Line:

Success on this problem requires systematically testing factor pairs and remembering that quadratic equations generally yield two solutions. The key insight is that finding factors isn't just about trial and error—there's a logical approach to narrowing down the possibilities based on the signs and magnitudes of the target product and sum.