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# Can every quadratic equation be factored?

try to solve it for factoring, then checking your answer using any other method. If you can't factor it but you can solve it using another method, how do you put the equation in its factored form.
24X²-14X-3 = 0

Hi Harry,
This equation can be solved by factoring.

When i am doing these, i like to think of all the possible factors of the coefficient in front of x^2.....in this case 24.
Here is what they could be:
1 x 24
2 x 12
3 x 8
4 x 6

then, i do the same for the constant (3 in this case). Since 3 is a prime number, there is only one set of factors:
1 x 3

now the "hard" part is choosing which factors to use to get the middle coefficient (-14 in this case)
Since it is negative, we know that we will need a + and a - in the two factors:

(?x + ?)(?x - ?)

a little experimenting and we can get:

(6x + 1)(4x - 3) = 0

If we generalize this equation to be A * B = 0. So if we make A = 0 then B doesn't matter because 0 * anything = 0.  Same idea for B. So what we do next is to make each factor a little mini equation like this:

(6x + 1) = 0
6x = -1
x = -1/6

(4x-3) = 0
4x = 3
x = 3/4

so the roots are:
x=3/4 or x = -1/6

Does that make sense?

Now, the harder part.....can every quadratic equation be factored? I believe the short answer is yes, but it may be very difficult if the coefficients are not easy numbers. However, there are different methods that you could use to figure it out (like completing the square and the quadratic formula ).  These methods are not as simple, but will work regardless of how "easy" the factors are.
aX2 + bx +c can be factored to , to binomial of ( X1 -a ) ( X2-b) , if and and only if

X+  X= -b/a
X 1 * X2  = c/a          a, b, c   be all integers

Every quadratic can technically be factored, but a lot of them can't be factored easily and not by the human brain without prior work.  Like x2 + 5x -7 = 0.  This can be factored like (x  +     )(x   -    ) = 0 but almost no one can just look at it and know what would go after the  + and the minus.  You would have to us the quadratic formula or complete the square and solve for x.  Then the answers you get can go in the blanks, but by then there isn't a point, since the goal of factoring is to solve for x, and you already did with the quad.form or completing the square.