To solve the problem of finding an equivalent expression to [tex]\( 2x^2 - 14x + 24 \)[/tex], we need to factorize the quadratic expression. Here is a step-by-step explanation of how we arrived at the correct factorization:
1. Identify and write down the quadratic expression:
[tex]\[
2x^2 - 14x + 24
\][/tex]
2. Factor out the greatest common factor (GCF) if there is one:
In this case, the GCF is 2.
[tex]\[
2(x^2 - 7x + 12)
\][/tex]
3. Factor the quadratic expression inside the parentheses [tex]\(x^2 - 7x + 12\)[/tex]:
To factor this, we need to find two numbers that multiply to give the constant term [tex]\(+12\)[/tex] and add to give the middle coefficient [tex]\(-7\)[/tex]. These two numbers are [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex].
4. Rewrite the quadratic expression using these factors:
[tex]\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\][/tex]
5. Combine this with the GCF:
[tex]\[
2(x - 3)(x - 4)
\][/tex]
Therefore, the factored form of [tex]\(2x^2 - 14x + 24\)[/tex] is:
[tex]\[
2(x - 3)(x - 4)
\][/tex]
Looking at the given multiple-choice options, we can see that the correct answer is:
B. [tex]\[ 2(x-3)(x-4) \][/tex]
