To determine which expression is equivalent to [tex]\( 2x^2 - 14x + 24 \)[/tex], we need to factor the quadratic expression. Follow these steps:
1. Identify the quadratic expression:
[tex]\[
2x^2 - 14x + 24
\][/tex]
2. Factor out the greatest common factor (GCF):
Observe that each term in the quadratic expression is divisible by 2. Factor out the 2:
[tex]\[
2(x^2 - 7x + 12)
\][/tex]
3. Factor the quadratic expression inside the parentheses [tex]\((x^2 - 7x + 12)\)[/tex]:
We need to find two numbers that multiply to [tex]\(12\)[/tex] (the constant term) and add up to [tex]\(-7\)[/tex] (the coefficient of the linear term). These numbers are [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex] because:
[tex]\[
-3 \times -4 = 12 \quad \text{and} \quad -3 + (-4) = -7
\][/tex]
4. Write the factored form:
Using the numbers [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex], the quadratic expression [tex]\(x^2 - 7x + 12\)[/tex] can be factored as:
[tex]\[
(x - 3)(x - 4)
\][/tex]
5. Include the factored out 2 from step 2:
Finally, we combine the GCF with the factored quadratic expression:
[tex]\[
2(x - 3)(x - 4)
\][/tex]
6. Identify the correct option:
Comparing this with the given options, we get:
[tex]\[
2(x - 3)(x - 4) \quad \text{is option C}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(2(x-3)(x-4)\)[/tex]
