SOLUTION: need help in understanding ((2a^5)/3b^-3))^4 and solving 24x^3-14x^2=90x

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Question 763356: need help in understanding
((2a^5)/3b^-3))^4
and solving
24x^3-14x^2=90x
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
%28%282a%5E5%29%2F3b%5E-3%29%29%5E4
= %282a%5E5b%5E3%2F3%29%5E4
= 16a%5E20b%5E12%2F81

and solving
24x%5E3-14x%5E2=90x
24x%5E3+-+14x%5E2+-+90x+=+0
2x%2A%2812x%5E2+-+7x+-+45%29+=+0
x = 0 *** Answer 1
---------
12x%5E2+-+7x+-+45+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 12x%5E2%2B-7x%2B-45+=+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=%28-7%29%5E2-4%2A12%2A-45=2209.

Discriminant d=2209 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--7%2B-sqrt%28+2209+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-7%29%2Bsqrt%28+2209+%29%29%2F2%5C12+=+2.25
x%5B2%5D+=+%28-%28-7%29-sqrt%28+2209+%29%29%2F2%5C12+=+-1.66666666666667

Quadratic expression 12x%5E2%2B-7x%2B-45 can be factored:
12x%5E2%2B-7x%2B-45+=+%28x-2.25%29%2A%28x--1.66666666666667%29
Again, the answer is: 2.25, -1.66666666666667. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+12%2Ax%5E2%2B-7%2Ax%2B-45+%29