You can put this solution on YOUR website! This is fun to factor completely
It is going to be a product of two linear terms
(ax + b) (cx + d)
For a and c I think it is going to be 1 and 3, but not sure which is which
For b and d it has to multiply to 40 so the choices are
1*40, 2*20, 4*10, or 5*8
Since the middle term is a minus I think b and d are going to be negative, or you can write it
(x - b) (3x - d)
How will it work out so the middle term is -23
3b + d = 23
I bet it is the 5 and 8 one
3*5 + 8 = 23
(x - 5)(3x - 8) = 3x^2 -15x -8x + 40
3x^2 - 23x + 40 = 0
That is how I learned to do it before I knew Completing the Square and the Quadratic Formula.
Completing the Square is a clearer process but you have to fool around with fractions
3x^2 - 23x + 40 = 0
x^2 -23/3 x + 40/3 = 0
On the left I want to see something that looks like x^2 + 2Cx + C^2, so I determine that C is -23/6. I add (-23/6)^2 to both sides to have this come out right. I move the 40/3 to the right so I can solve for the squared term on the left.
x^2 + 2(-23/6) x + (-23/6)^2 = -40/3 + (-23/6)^2
There I added the term I will need, (-23/6)^2, to both sides
in order to have the left side be a perfect square. Notice
how the left side factors now:
(x - 23/6)^2 = -40/3 + 529/36 = (529 - 480)/36 = (49/36)
x - 23/6 = 7/6 or -7/6
x = (23+7)/6 = 5
or x = (23 - 7)/6 = 16/6
Therefore again the answer is 3(x-5)(x-8/3) or (x-5)(3x-8).
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