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Select the correct answer.

Which expression is equivalent to the given expression?

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

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

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

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

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

Sagot :

GinnyAnsweridn
To determine which expression is equivalent to [tex]\(2x^2 - 14x + 24\)[/tex], we can try to factorize it.

Let's factorize the quadratic expression:

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

We need to find two numbers that multiply to [tex]\(2 \times 24 = 48\)[/tex] and add up to [tex]\(-14\)[/tex].

The pairs that multiply to 48 are:
- [tex]\(1 \times 48\)[/tex]
- [tex]\(2 \times 24\)[/tex]
- [tex]\(3 \times 16\)[/tex]
- [tex]\(4 \times 12\)[/tex]
- [tex]\(6 \times 8\)[/tex]

When considering their sums and differences, none of these pairs add up to [tex]\(-14\)[/tex] except for [tex]\((-6)\)[/tex] and [tex]\((-8)\)[/tex]:

[tex]\(-6 \times -8 = 48\)[/tex]

[tex]\(-6 + -8 = -14\)[/tex]

Hence, we split the middle term and factor by grouping:

[tex]\[2x^2 - 6x - 8x + 24\][/tex]

Now, group the terms:

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

This can be factored as:

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

But we see [tex]\(2x - 8\)[/tex] can be further factored as [tex]\(2(x - 4)\)[/tex]:

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

Therefore, the factored form of [tex]\(2x^2 - 14x + 24\)[/tex] is:

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

Thus, the correct answer is:

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