How do you factor 6a^2 + 17ab + 12b^2
You just have to guess-and-check for factoring. (After guessing, FOIL to make sure that it works out.) Typically, the numbers will be the closest factors of the number. For example, 12 = 4*3. (Those are the closest factors.) 6 = 3*2. (Those are the closest factors.) Here you go...
(3a+4b)(2a+3b)
To factor the expression 6a^2 + 17ab + 12b^2, we need to look for two binomials that multiply together to give us the original expression. The first step is to break down the middle term (the term with "ab") by finding two numbers that multiply to give the product of the first and last term (the term with "6a^2" and "12b^2") and add up to the coefficient of the middle term (the coefficient of "ab").
In this case, the product of the first and last term is (6a^2)(12b^2) = 72a^2b^2. Now let's find two numbers that multiply to give us 72 and add up to 17. The numbers that satisfy this condition are 9 and 8.
Next, we rewrite the expression 17ab as 9ab + 8ab. Now we can factor by grouping:
6a^2 + 9ab + 8ab + 12b^2
Factor by grouping:
(6a^2 + 9ab) + (8ab + 12b^2)
Now we can factor out the greatest common factor from each grouping:
3a(2a + 3b) + 4b(2a + 3b)
Finally, notice that we have a common binomial term: (2a + 3b). We can factor it out of both terms:
(2a + 3b)(3a + 4b)
Therefore, the factored form of the expression 6a^2 + 17ab + 12b^2 is (2a + 3b)(3a + 4b).