SOLUTION: Solve using the quadratic equation {{{3x^2+2.5x=12.5}}}

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Question 857661: Solve using the quadratic equation 3x%5E2%2B2.5x=12.5
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
3x^2+2.5x-12.5=0
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 6%2Ax%5E2%2B5%2Ax-25=0 ( notice a=6, b=5, and c=-25)





x+=+%28-5+%2B-+sqrt%28+%285%29%5E2-4%2A6%2A-25+%29%29%2F%282%2A6%29 Plug in a=6, b=5, and c=-25




x+=+%28-5+%2B-+sqrt%28+25-4%2A6%2A-25+%29%29%2F%282%2A6%29 Square 5 to get 25




x+=+%28-5+%2B-+sqrt%28+25%2B600+%29%29%2F%282%2A6%29 Multiply -4%2A-25%2A6 to get 600




x+=+%28-5+%2B-+sqrt%28+625+%29%29%2F%282%2A6%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-5+%2B-+25%29%2F%282%2A6%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-5+%2B-+25%29%2F12 Multiply 2 and 6 to get 12


So now the expression breaks down into two parts


x+=+%28-5+%2B+25%29%2F12 or x+=+%28-5+-+25%29%2F12


Lets look at the first part:


x=%28-5+%2B+25%29%2F12


x=20%2F12 Add the terms in the numerator

x=5%2F3 Divide


So one answer is

x=5%2F3




Now lets look at the second part:


x=%28-5+-+25%29%2F12


x=-30%2F12 Subtract the terms in the numerator

x=-5%2F2 Divide


So another answer is

x=-5%2F2


So our solutions are:

x=5%2F3 or x=-5%2F2


Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B2.5x%2B-12.5+=+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=%282.5%29%5E2-4%2A3%2A-12.5=156.25.

Discriminant d=156.25 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2.5%2B-sqrt%28+156.25+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282.5%29%2Bsqrt%28+156.25+%29%29%2F2%5C3+=+1.66666666666667
x%5B2%5D+=+%28-%282.5%29-sqrt%28+156.25+%29%29%2F2%5C3+=+-2.5

Quadratic expression 3x%5E2%2B2.5x%2B-12.5 can be factored:
3x%5E2%2B2.5x%2B-12.5+=+3%28x-1.66666666666667%29%2A%28x--2.5%29
Again, the answer is: 1.66666666666667, -2.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B2.5%2Ax%2B-12.5+%29