Factor by Grouping.
15r^2-19r+6
multiply 15 by 6 to get 90
Now find two numbers which when multiplied will give you +90, and when added will give you -19
I found -10 and -9
so replace the -19r with -10r - 9r
15r^2-19r+6
= 15r^2 - 10r - 9r + 6
Here is where the "grouping" comes in
= 5r(3r - 2) - 3(3r - 2)
= (3r - 2)(5r - 3)
To factor by grouping, we'll look for common factors within the terms and group them together.
Given the quadratic expression: 15r^2 - 19r + 6
First, let's find two numbers whose product is equal to the product of the coefficients of the first and last term, which is 15 * 6 = 90, and whose sum is equal to the coefficient of the middle term, which is -19.
The numbers that meet these conditions are -10 and -9 since (-10) * (-9) = 90 and (-10) + (-9) = -19.
Next, we'll rewrite the middle term -19r using the numbers -10 and -9 as follows:
15r^2 - 10r - 9r + 6
Now, we group the terms into two pairs, and factor out the greatest common factor from each pair:
(15r^2 - 10r) + (-9r + 6)
We can factor out a common factor of 5r from the first pair, and a common factor of -3 from the second pair:
5r(3r - 2) - 3(3r - 2)
Notice that both pairs have a common binomial factor, which is (3r - 2). We'll factor out this binomial factor:
(3r - 2)(5r - 3)
Therefore, the factored form of 15r^2 - 19r + 6 is (3r - 2)(5r - 3).