SOLUTION: Solve the following: x^3 + 2x^2 = -5x

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Question 113859: Solve the following: x^3 + 2x^2 = -5x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E3%2B2x%5E2=-5x Start with the given equation


x%5E3%2B2x%5E2%2B5x=0 Move all of the terms to the left side




x%28x%5E2%2B2x%2B5%29=0 Factor out the GCF x


x=0 or x%5E2%2B2x%2B5=0 Set each factor equal to zero

So this means we have one solution of x=0

Now let's focus on the second equation x%5E2%2B2x%2B5=0


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 x%5E2%2B2%2Ax%2B5=0 ( notice a=1, b=2, and c=5)




x+=+%28-2+%2B-+sqrt%28+%282%29%5E2-4%2A1%2A5+%29%29%2F%282%2A1%29 Plug in a=1, b=2, and c=5



x+=+%28-2+%2B-+sqrt%28+4-4%2A1%2A5+%29%29%2F%282%2A1%29 Square 2 to get 4



x+=+%28-2+%2B-+sqrt%28+4%2B-20+%29%29%2F%282%2A1%29 Multiply -4%2A5%2A1 to get -20



x+=+%28-2+%2B-+sqrt%28+-16+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-2+%2B-+4%2Ai%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-2+%2B-+4%2Ai%29%2F%282%29 Multiply 2 and 1 to get 2



After simplifying, the quadratic has roots of

x=-1+%2B+2%2Ai or x=-1+-+2%2Ai

Notice if we graph the quadratic y=x%5E2%2B2%2Ax%2B5, we get

+graph%28+500%2C+500%2C+-16%2C+14%2C+-11%2C+19%2C+x%5E2%2B2%2Ax%2B5%29+ graph of y=x%5E2%2B2%2Ax%2B5

And we can see that there are no real roots

To visually verify the answer, check out this page to see a visual representation of imaginary roots


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Answer:


So we have the solutions : x=0, x=-1+%2B+2%2Ai or x=-1+-+2%2Ai