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Factor completely [tex]$2x^2 - 16x + 30$[/tex].

A. [tex]$2(x+3)(x+5)$[/tex]
B. [tex][tex]$2(x-3)(x+5)$[/tex][/tex]
C. [tex]$2(x-5)(x+3)$[/tex]
D. [tex]$2(x-5)(x-3)$[/tex]


Sagot :

To factor the quadratic expression [tex]\(2x^2 - 16x + 30\)[/tex] completely, we will proceed through the following steps:

1. Identify a common factor if possible:
The given expression is [tex]\(2x^2 - 16x + 30\)[/tex].
- Here, we notice that each term in the quadratic expression contains a factor of 2. So, we can start by factoring out 2 from the entire expression.
[tex]\[ 2(x^2 - 8x + 15) \][/tex]

2. Factor the quadratic expression inside the parentheses:
Now, we need to focus on factoring [tex]\(x^2 - 8x + 15\)[/tex]. We look for two numbers that multiply to give the constant term (15) and add to give the coefficient of the middle term (-8).

- List down the pair of factors of 15 and find the pair that adds up to -8:
- [tex]\(1 \times 15 = 15\)[/tex] and [tex]\(1 + 15 = 16\)[/tex]
- [tex]\(3 \times 5 = 15\)[/tex] and [tex]\(3 + 5 = 8\)[/tex]
- Since we need the sum to be negative, consider the factors with negative signs: [tex]\((-3) \times (-5) = 15\)[/tex] and [tex]\((-3) + (-5) = -8\)[/tex].

Hence, the two numbers are -3 and -5.

3. Write the factored form:
Utilizing the numbers we found, we can now factor the quadratic expression:
[tex]\[ x^2 - 8x + 15 = (x - 3)(x - 5) \][/tex]

4. Combine with the factor we initially factored out:
Now, we multiply back the factor of 2 that we had factored out earlier:
[tex]\[ 2(x - 3)(x - 5) \][/tex]

Thus, the completely factored form of the given quadratic expression [tex]\(2x^2 - 16x + 30\)[/tex] is:
[tex]\[ 2(x - 3)(x - 5) \][/tex]

Therefore, the correct answer is:
[tex]\[ 2(x - 5)(x - 3) \][/tex]
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