Question

Find the solution(s) to the quadratic equation x^2 − 10x + 24, and then select the correct answer from the following options.
A) Only x = 4 is a solution.
B) Neither x = 4 nor x = 6 is a solution.
C) Both x = 4 and x = 6 are solutions.
D) Only x = 6 is a solution.

Answer 1

To find the solutions of the quadratic equation \(x^2 - 10x + 24 = 0\), we can use the factoring method or the quadratic formula. Let's try to factor it first.

We need two numbers that multiply to \(24\) (the constant term) and add up to \(-10\) (the coefficient of the linear term).

The numbers \(-4\) and \(-6\) fit this criterion:
\[
-4 \times -6 = 24 \quad \text{and} \quad -4 + -6 = -10
\]

This means we can factor the quadratic as:
\[
(x - 4)(x - 6) = 0
\]

Setting each factor equal to zero gives us the solutions:
1. \(x - 4 = 0\) leads to \(x = 4\)
2. \(x - 6 = 0\) leads to \(x = 6\)

Thus, the two solutions to the equation \(x^2 - 10x + 24 = 0\) are \(x = 4\) and \(x = 6\).

Now, looking at the available options:
A) Only x = 4 is a solution.
B) Neither x = 4 nor x = 6 is a solution.
C) Both x = 4 and x = 6 are solutions.
D) Only x = 6 is a solution.

The correct answer is:
C) Both x = 4 and x = 6 are solutions.