To factor the expression [tex]\( 18x^2 - 60x + 50 \)[/tex] completely, follow these steps:
1. Identify and extract the Greatest Common Factor (GCF).
- The coefficients of the terms are 18, -60, and 50. The GCF of 18, -60, and 50 is 2.
- Factoring out 2 from each term gives us:
[tex]\[
18x^2 - 60x + 50 = 2 \left( 9x^2 - 30x + 25 \right)
\][/tex]
2. Factor the quadratic expression inside the parentheses:
- We need to factor [tex]\( 9x^2 - 30x + 25 \)[/tex].
3. Examine the quadratic polynomial [tex]\( 9x^2 - 30x + 25 \)[/tex]:
- We look for factors of [tex]\( a \cdot c = 9 \cdot 25 = 225 \)[/tex], which add up to [tex]\( b = -30 \)[/tex].
4. Find the factors of 225 that add to -30:
- The factors are -15 and -15. So, we can split the middle term as follows:
[tex]\[
9x^2 - 30x + 25 = 9x^2 - 15x - 15x + 25
\][/tex]
5. Group the terms and factor by grouping:
- Grouping the terms gives:
[tex]\[
(9x^2 - 15x) + (-15x + 25)
\][/tex]
- Factoring out the common factors from each group:
[tex]\[
3x(3x - 5) - 5(3x - 5)
\][/tex]
6. Factor out the common binomial [tex]\( (3x - 5) \)[/tex]:
- This gives us:
[tex]\[
3x(3x - 5) - 5(3x - 5) = (3x - 5)(3x - 5) = (3x - 5)^2
\][/tex]
7. Substitute back into the original expression:
- Recall we factored out a 2 initially. Combining everything gives:
[tex]\[
18x^2 - 60x + 50 = 2(3x - 5)^2
\][/tex]
8. Conclude the complete factored form:
- The final factored form is:
[tex]\[
\boxed{2(3x - 5)^2}
\][/tex]
Therefore, the correct answer is [tex]\( \boxed{2(3x - 5)^2} \)[/tex], which matches option C.
