SOLUTION: Solve the equation 1/3 x^2 + 1/2x = 1/3

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Question 162302: Solve the equation
1/3 x^2 + 1/2x = 1/3
Found 2 solutions by ankor@dixie-net.com, Alan3354:
Answer by ankor@dixie-net.com(15649) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the equation
%281%2F3%29x%5E2 + %281%2F2%29x = 1%2F3
:
We can get rid of those annoying denominator by multiplying by 6:
6*%281%2F3%29x%5E2 + 6*%281%2F2%29x = 6*1%2F3
Cancel out the denominators and you have:
2x^2 + 3x = 2
:
Arrange as a quadratic equation:
2x^2 + 3x - 2 = 0
:
This factors to:
(2x - 1)(x + 2) = 0
:
2x = +1
x = 1%2F2
and
x = -2
:
Check x=-2 in original equation
%281%2F3%29%28-2%5E2%29 + %281%2F2%29%28-2%29 = 1%2F3
%281%2F3%29%284%29 + %28%28-2%29%2F2%29 = 1%2F3
4%2F3 - 1 = 1%2F3
:
I give you the honor of checking the x = 1/2 solution

Answer by Alan3354(30993) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the equation
1/3 x^2 + 1/2x = 1/3
---------------
Multiply by 6
2x^2 + 3x = 2
2x^2 + 3x - 2 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B3x%2B-2+=+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=%283%29%5E2-4%2A2%2A-2=25.

Discriminant d=25 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+25+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+25+%29%29%2F2%5C2+=+0.5
x%5B2%5D+=+%28-%283%29-sqrt%28+25+%29%29%2F2%5C2+=+-2

Quadratic expression 2x%5E2%2B3x%2B-2 can be factored:
2x%5E2%2B3x%2B-2+=+%28x-0.5%29%2A%28x--2%29
Again, the answer is: 0.5, -2. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B3%2Ax%2B-2+%29

The onsite solver gets the right answers, but the factors are not right when the coefficient of the x^2 term is not 1.