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]
