Select the correct answer.

Which expression is equivalent to the given expression?

[tex]\[2x^2 - 14x + 24\][/tex]

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]



Answer :

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]