Which expression is equivalent to the given expression?

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

A. [tex]\((2x - 12)(x - 2)\)[/tex]

B. [tex]\(2(x - 8)(x + 3)\)[/tex]

C. [tex]\(2(x - 5)(x - 2)\)[/tex]

D. [tex]\(2(x - 3)(x - 4)\)[/tex]



Answer :

To determine which expression is equivalent to the given expression [tex]\(2x^2 - 14x + 24\)[/tex], let's go through the process of expanding each of the provided choices and compare them to the original expression.

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

Choices:

A. [tex]\((2x - 12)(x - 2)\)[/tex]

First, we expand this expression:
[tex]\[ (2x - 12)(x - 2) \\ = 2x(x - 2) - 12(x - 2) \\ = 2x^2 - 4x - 12x + 24 \\ = 2x^2 - 16x + 24 \][/tex]

Clearly, the expanded form [tex]\(2x^2 - 16x + 24\)[/tex] does not match the given expression.

B. [tex]\(2(x - 8)(x + 3)\)[/tex]

Next, we expand this expression:
[tex]\[ 2(x - 8)(x + 3) \\ = 2[(x - 8)(x + 3)] \\ = 2[x^2 + 3x - 8x - 24] \\ = 2[x^2 - 5x - 24] \\ = 2x^2 - 10x - 48 \][/tex]

The expanded form [tex]\(2x^2 - 10x - 48\)[/tex] does not match the given expression.

C. [tex]\(2(x - 5)(x - 2)\)[/tex]

Next, we expand this expression:
[tex]\[ 2(x - 5)(x - 2) \\ = 2[(x - 5)(x - 2)] \\ = 2[x^2 - 2x - 5x + 10] \\ = 2[x^2 - 7x + 10] \\ = 2x^2 - 14x + 20 \][/tex]

The expanded form [tex]\(2x^2 - 14x + 20\)[/tex] does not match the given expression.

D. [tex]\(2(x - 3)(x - 4)\)[/tex]

Finally, we expand this expression:
[tex]\[ 2(x - 3)(x - 4) \\ = 2[(x - 3)(x - 4)] \\ = 2[x^2 - 4x - 3x + 12] \\ = 2[x^2 - 7x + 12] \\ = 2x^2 - 14x + 24 \][/tex]

The expanded form [tex]\(2x^2 - 14x + 24\)[/tex] matches the given expression exactly.

Thus, the expression that is equivalent to [tex]\(2x^2 - 14x + 24\)[/tex] is choice:

D. [tex]\(2(x - 3)(x - 4)\)[/tex]