Question

How do you factor and solve `6x^2 - 5x = 6`?

Answer 1

`x=-2/3` or `x=3/2`

Explanation:

First subtract `6` from both sides to get:

`6x^2-5x-6 = 0`

Factor using an AC method:

Find a pair of factors of `AC = 6*6=36` which differ by `B=5`.

The pair `9, 4` works. Use this pair to split the middle term, then factor by grouping:

`0 = 6x^2-5x-6`

`=6x^2-9x+4x-6`

`=(6x^2-9x)+(4x-6)`

`=3x(2x-3)+2(2x-3)`

`=(3x+2)(2x-3)`

Hence zeros: `x=-2/3` and `x=3/2`

Answer 2

The solutions are the values of `x` that satisfy `6x^2-5x-6=0`

Explanation:

By completing the square you can get the formula for the solutions (roots) of a 2nd degree polynomial equation `ax^2+bx+c=0`, as follows `x_1,_2 = (-b+-sqrt(b^2-4ac))/(2a)`. In this case we have `a=6`, `b=-5` and `c=-6`

So the first solution is `x_1 = (-b+sqrt(b^2-4ac))/(2a)=(5+sqrt(25-4*6*(-6)))/(2*6)=(5+sqrt(25+144))/(12)=(5+sqrt(169))/(12)=(5+13)/12=18/12=3/2`

Similarly, the second solution is `(5-13)/12=-8/12=-2/3`