SOLUTION: How do I solve an equation by factoring, if the sum in a fraction? EXAMPLE: 6x + 7 = 20/x

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: How do I solve an equation by factoring, if the sum in a fraction? EXAMPLE: 6x + 7 = 20/x      Log On


   

Question 154390: How do I solve an equation by factoring, if the sum in a fraction?
EXAMPLE: 6x + 7 = 20/x
Found 3 solutions by Alan3354, scott8148, ankor@dixie-net.com:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
How do I solve an equation by factoring, if the sum in a fraction?
EXAMPLE: 6x + 7 = 20/x
------------
Multiply thru by x
6x^2 + 7x = 20
6x^2 + 7x - 20 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B7x%2B-20+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%287%29%5E2-4%2A6%2A-20=529.

Discriminant d=529 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+529+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%287%29%2Bsqrt%28+529+%29%29%2F2%5C6+=+1.33333333333333
x%5B2%5D+=+%28-%287%29-sqrt%28+529+%29%29%2F2%5C6+=+-2.5

Quadratic expression 6x%5E2%2B7x%2B-20 can be factored:
6x%5E2%2B7x%2B-20+=+%28x-1.33333333333333%29%2A%28x--2.5%29
Again, the answer is: 1.33333333333333, -2.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B7%2Ax%2B-20+%29

Once you know the zeros, you can multiply by 2 and 3 to make them integers and figure out how to factor it.
The alternative is trial and error.
(3x-4)*(2x+5)

Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
multiplying by x __ 6x^2+7x=20 __ subtracting 20 __ 6x^2+7x-20=0

factoring __(2x+5)(3x-4)=0

2x+5=0 __ x=-5/2

3x-4=0 __ x=4/3

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
6x + 7 = 20/x
;
Multiply both sides by x to get rid of the denominator
x(6x + 7) = 20
:
Mult terms inside the brackets:
6x^2 + 7x = 20
;
Subtract 20 from both sides to reveal our old friend, the quadratic equation
6x^2 + 7x - 20 = 0
:
With diligent effort we can factor this to:
(2x + 5)(3x - 4) = 0
Two solutions
2x = -5
x = -5%2F2
x = -2.5
and
3x = 4
x = 4%2F3
;
:
Check the 1st solution in the original equation
6(-2.5) + 7 = 20%2F-2.5
-15 + 7 = -8
;
Check the 2nd solution
6*4%2F3 + 7 = 20%283%2F4%29
8 + 7 = 15