How do you factor this out? 6x^2+19x+7=90?

Question

Please provide step by step and show how you work this out by the method of (x )(x ).

Answer 1

First, move the 90 to the left side of the equals sign.

6x^2 + 19x -83 = 0

83 is prime, so the only way to get 83 is 1*83.
The x^2 coefficient is 6... that can be gotten as 1*6 or 2*3.
There's no way of combining these possibilities in a way that gives a coefficient of 19 for the x term.

The best we can do is something like (6x+83)*(x-1) = 6x^2 +77x -83, but there's no way to get a 19 in there.

Accordingly, this particular equation is not factorable.

Answer 2

7 is a prime number, but 6 = 3 x 2, so we'll have two possibilities to start with:

(1) (3x +/- 7)(2x -/+ 1) and
(2) (2x +/- 7)(3x -/+ 1)

(There are two other possibilities, involving 6x and x, but let's try these first.)

Do FOIL -- outside minus inside -- on (1) and (2):

For (1): 3*1 - 7*2 = 3 - 14 = 11 (sign doesn't matter)(no good)
For (2): 2*1 - 7*3 = 2 - 21 = 19 (this one works!)

So your answer, from (2) is:

(2x + 7)((3x - 1) = 90

That's how you factor it, step by step. Notice the signs ... when you multiply this out using FOIL, you get

6x^2 - 2x + 21x - 7 = 90
6x^2 + 19x - 7 = 90

The answer checks.

Answer 3

6x^2+19x-83 = 6(x-x1)(x-x2)

where x1 and x2 are the roots of the quadratic x^2+19/6x-83/6