To determine which of the given options is equivalent to the expression [tex]\(2x^2 - 14x + 24\)[/tex], we need to factorize the quadratic expression. Here are the steps for factorizing and verifying the correct form:
1. Write down the given expression:
[tex]\[
2x^2 - 14x + 24
\][/tex]
2. Factor out the common factor from the entire expression:
[tex]\[
2(x^2 - 7x + 12)
\][/tex]
3. Factorize the quadratic expression inside the parentheses:
We need to find two numbers whose product is [tex]\(12\)[/tex] (the constant term) and whose sum is [tex]\(-7\)[/tex] (the coefficient of the middle term).
These numbers are [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex], because:
[tex]\[
-3 \times -4 = 12
\][/tex]
[tex]\[
-3 + -4 = -7
\][/tex]
4. Write the factorized form of the quadratic expression:
[tex]\[
x^2 - 7x + 12 = (x - 3)(x - 4)
\][/tex]
5. Include the common factor we factored out initially:
[tex]\[
2(x - 3)(x - 4)
\][/tex]
Therefore, the expression [tex]\(2x^2 - 14x + 24\)[/tex] factorizes to:
[tex]\[
2(x - 3)(x - 4)
\][/tex]
6. Match the factorized form with the given options:
A. [tex]\(2(x - 3)(x - 4)\)[/tex]
B. [tex]\(2(x - 5)(x - 2)\)[/tex]
C. [tex]\(2(x - 8)(x + 3)\)[/tex]
D. [tex]\((2x - 12)(x - 2)\)[/tex]
After comparing, it is clear that option A matches our factorized result.
Hence, the correct answer is:
A. [tex]\(2(x - 3)(x - 4)\)[/tex]
