Find the greatest common factor of 12x^2+18x-36
6(2x^2 + 3x - 6)
To find the greatest common factor (GCF) of the given expression 12x^2 + 18x - 36, we need to factorize the expression completely.
First, let's factor out the greatest common factor of the coefficients, which is 6:
6(2x^2 + 3x - 6)
Now let's focus on factoring the quadratic expression inside the parentheses, 2x^2 + 3x - 6. We can do this by grouping:
2x^2 + 3x - 6 = (2x^2 + 6x) + (-3x - 6)
Now let's factor out the greatest common factor from each group:
2x(x + 3) - 3(x + 3)
Notice that (x + 3) is a common factor in both terms. Let's factor it out:
(x + 3)(2x - 3)
Now we can see that the completely factorized expression is:
6(x + 3)(2x - 3)
So, the greatest common factor of 12x^2 + 18x - 36 is 6(x + 3)(2x - 3).
To find the greatest common factor of the expression 12x^2 + 18x - 36, we need to factorize the expression and find the highest exponent for each common factor.
Step 1: Look for common factors. In this case, all the coefficients (12, 18, and 36) share a common factor, which is 6.
12x^2 + 18x - 36 = 6(2x^2 + 3x - 6)
Step 2: Now we need to factorize the quadratic expression 2x^2 + 3x - 6. To do this, we can either use factoring methods or the quadratic formula. Let's try factoring.
2x^2 + 3x - 6 = (2x - 3)(x + 2)
Step 3: Now let's put everything together and find the common factors.
12x^2 + 18x - 36 = 6(2x^2 + 3x - 6) = 6(2x - 3)(x + 2)
So, the greatest common factor of 12x^2 + 18x - 36 is 6(2x - 3)(x + 2).