SOLUTION: 3x�-31x-60=0 plz explain me in detail

Question 1014361: 3x�-31x-60=0 plz explain me in detail
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
is a quadratic equation.
That means that it can be written (in this case, it is written)
as a degree 2 polynomial equal to zero.
In simpler words, appears squared as in the equation;
it may also appear as , maybe multiplied by a number,
and there is no other strange business with , no in a denominator, no , no , etc).

There are 3 possible methods to solve a quadratic equation:
factoring (which only works if there are rational solutions),
"completing the square", and
using the quadratic formula.

In this case, I would use the quadratic formula.
The quadratic formula says that the solutions to an equation of the form
, if they exist in the math class you are taking,
are given by
For <--> ,
, , and , so

That gives you two solutions that exist,
because the square root of a positive number exists for most math class levels.
In some cases those square roots are irrational numbers, such as ,
but in this case turned out to be a nice whole number.
So, <--> gives us two solutions
.

THE FACTORING WAY TO SOLVE IT
consists of figuring out that
and re-writing as .
If , one of those two bracketed factors must be zero, so
--->---> .
Figuring out that requires some thinking,
but you do not have to mess with the quadratic formula.
To begin, you multiply the coefficient of and the independent term (the one without );
.
Then you look for pairs of factors of that number
that add up to the coefficient of in your equation: .
You look for pairs of factors of , and worry about the minus sign later.
Sometimes there is a lot of pairs, and you have to organize.
I like to look for the smaller factor starting with ,
and then see if I can make factor pairs where the smaller factor is , , , etc, in that order.

and are not factors of

is not a factor of 180

and are not factors of
and factors equal to and larger have already been found.
From the list above, we know that
and are factor pairs that multiply to yield ,
but you want them to add up to .
The sum has the wrong sign.
has the expected negative sign, and is close to .
Obviously we want to give the minus sign to the larger factor, to make the sum negative.
You also see that and are a little to far apart,
and that makes the difference larger than .
So we try a pair of factors that are a little closer together:
and ,
and discover that we have found our pair of factors.
Now, you re-write using those factors, like this:
.
Finally, find common factors, and "take out the common factors", twice:

Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!

Since the leading coefficient 3 can only be factored only one way, I think the trial-and-error factoring method is better in this case: 3x�-31x-60=0 Since the last term -60 is negative, we know that the absolute value of the middle coefficient, -31, which is 31, must be the result when the absolute values of the product of the outer and inner terms are subtracted. [If the last term were +, we would know that they would be added]. The factorizations are either of these forms: (x A)(3x B) (x B)(3x A) where A and B are as in this chart: ∥ A, B ∥ |B-3A|,|A-3B| ∥ 1,60 ∥ 57 179 ∥ 2,30 ∥ 24 88 ∥ 3,20 ∥ 11 57 ∥ 4,15 ∥ 3 41 ∥ 5,12 ∥ 3 31 ∥ 6,10 ∥ 8 24 How to make that chart: In the first column we list all the factor pairs of 60. Then we form the differences mentally, until we come to the absolute value of the coefficient of the middle term, 31. In this case it was the red one. Often when using this method, we will reach it much sooner, and stop whenever we reach it. Now since the 31 came from the second column we know that the factorization is of the form: (x B)(3x A) so the factorization is (x 12)(3x 5) Now the product of the outer terms is 5x and the product of the inner terms is 36x, so to get the middle term -31x, the 36x must be made negative and the 5x must be made positive. So the factorization is (x-12)(3x+5). It takes some practice to use this method, but it'll save time when either the first or last term can be factored only one way. Edwin

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