How can you factor this?

Question

6x^2 - x - 2 = 0

Can you please just explain how to factor this? I already know the answers from doing the first part of the question. Thank you.

How did you get the -4x + 3x?

Answer 1

it's like this: first, do the "ac test." this means, you must multiply first the numerical coefficient of the first and last terms of the equation. The numerical coefficients of the first and last terms of the equation are: 6 and -2. their product is -12. next, you must find a pair of factors of the product such that when added up, will equal to the numerical coefficient of the MIDDLE term, which is -1. first, let's list down all the factors of -12 and their sums:
-1 and 12 = 11
1 and -12 = -11
-2 and 6 = 4
2 and -6 = -4
-3 and 4 = 1
3 and -4 = -1
it is clear from here that only the pair "3 and -4" give the sum of -1.
From here, you must expand the middle term "-x" to "-4x+3x".
the equation will now become:
6x^2 - 4x + 3x -2 = 0
Apply here the method of "factoring by grouping." you must group together the first 2 terms together and the last 2 terms likewise. the equation is now:
(6x^2 - 4x) + (3x - 2) = 0
Next, you must factor out the GREATEST COMMON FACTOR from each group, making the equation:
2x(3x - 2) + 1(3x - 2) = 0
From this stage, again factor out the GCF which is (3x - 2) and group together its coefficient in both terms. it will become:
(3x - 2)(2x + 1) = 0
Since you only asked for the process of factoring, I won't include anymore the answers.

P.S. The answers are: x = 2/3 or -1/2
;-)

Answer 2

I assume you can factor when the leading coefficient is a "1". For example, if you have x^2 - 2x - 15 = 0 then you see the leading coefficient is a 1. To find your factors you think of 2 numbers which multiply to equal the -15 and add to equal the -2. Since (-5)*(3) = -15 and (-5)+(3) = -2 your factors are (x-5)(x+3).

Use that same method to factor your problem, only since the leading coeffecient is 6, not 1, you need to multiply the 6 times the last number (-2) and you get (-12). Now think of numbers that multiply to equal -12 and add to get -1. That pair of numbers is (-4) and 3 because (-4)*(3)= -12 and (-4)+3= -1.
Rewrite the problem. Break up the middle term using the (-4) and 3:

6x^2 -4x +3x - 2 = 0

Now factor by grouping:
(6x^2 -4x) + (3x-2) = 0
2x (3x-2) + 1(3x-2) =0

Since the terms in parenthesis are the same, you can "undistribute) the 2x+1 and there are your factors:
(2x + 1)(3x -2)

This method will always work if your leading coefficient is not a 1.
For example:
12x^2 + 5x -3 =0.

Multiply the first and last coefficients: 12*(-3) = -36
Find 2 numbers that multiply to be -36 and add to be 5: 9 and (-4)
Rewrite the problem: 12x^2 +9x - 4x -3 = 0
Factor by grouping: (12x^2 +9x) + (-4x-3) = 0
Factor by grouping (cont): 3x(4x +3) -1(4x +3) =0
Since the terms in ( ) are the same: (3x - 1)(4x +3) =0

Answer 3

6x^2 - x - 2 = 0

Here's how I'd do it. For sure, the constant 2 is 2 times 1. No other choice. Next, I'd guess that the 6 might be 3 times 2, although it might be 6 times 1.

Using my first guess (6 = 3 x 2), I'd set it up this way:

(3x 1)(2x 2) (3x 2)(2x 1)

intentionally leaving the signs out.

Now take the difference of the outer product minus the inner product (from the FOIL method). In our first guess, it's 3 x 2 - 2 x 1 = 6 - 2 = 4. No good. (We want a 1.) In our second guess, it's 3 x 1 - 2 x 2 = 3 - 4 = -1. The sign doesn't matter here; we got what we needed!

Now we can write down the answer, inserting the sign to make the answer correct:

6x^2 - x - 2 = (3x - 2)(2x + 1)

In doing the signs here, notice that we have +3 - 4 = -1, not -3 + 4 = +1.

Answer 4

12x^2 +9x - 4x -3 = 0

this is another method of factoring where x^2 has a base of more than 1 (x^2 is the same as 1x^2). Notice that there are four terms in this equation. Usually you wouldn't see them in this form, this is presented for a learnign exercise, I think.

so you'd take the first two terms and see if there's any thing you can factor out of them, and there is:
12x^2 + 9x (yes, factor out like terms 3x)
3x(4x + 3)
now look at the last two terms:
-4x -3 (in this case you will factor out a -1)
-1(4x +3).
You need to see two matching terms inside the parentheses. These are the first term in you equation.
Now look at the individual numbers that are in front of the parentheses and put them together:
3x(4x+3)-1(4x+3) This becomes:
(4x+3)(3x-1) = 0
And you say you understand the rest...

Answer 5

6x^2-4x+3x-2=0
(2x+1) (3x-2)=0
so x=-1/2 or 2/3

Answer 6

reverse the FOIL process.....
6x^2-x-2=0

(2x+1) (3x-2)=0