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]
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]
