Sure, let's factor the quadratic expression [tex]\(2x^2 - 18x + 36\)[/tex] completely.
1. Identify the Greatest Common Factor (GCF):
First, we look for the greatest common factor among the terms. The coefficients are 2, -18, and 36. The GCF of these numbers is 2. Let's factor out the GCF from the expression:
[tex]\[
2x^2 - 18x + 36 = 2(x^2 - 9x + 18)
\][/tex]
2. Factor the Quadratic Expression:
Next, we need to factor the quadratic expression inside the parentheses: [tex]\(x^2 - 9x + 18\)[/tex].
To do this, we look for two numbers that multiply to 18 (the constant term) and add up to -9 (the coefficient of the linear term). These two numbers are -3 and -6 since:
[tex]\[
-3 \times -6 = 18
\][/tex]
[tex]\[
-3 + (-6) = -9
\][/tex]
3. Rewrite the Quadratic Expression:
Using the two numbers we found, we can rewrite the quadratic expression as the product of two binomials:
[tex]\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\][/tex]
4. Combine Our Results:
Now we substitute back the factored form of the quadratic expression into the original expression, bringing back the factored-out [tex]\(2\)[/tex]:
[tex]\[
2(x^2 - 9x + 18) = 2(x - 3)(x - 6)
\][/tex]
So, the completely factored form of [tex]\(2x^2 - 18x + 36\)[/tex] is:
[tex]\[
2(x - 3)(x - 6)
\][/tex]
