SOLUTION: 1. If x + a/x = b then find the value of (x^2 + bx + a)/(bx^2 - x^3) 2. If x + 1/x = 99, find the value of 100x/(3x^2 + 103x + 3) Question from Tata Mc Graw-Hill NTSE b

Algebra ->  Inverses -> SOLUTION: 1. If x + a/x = b then find the value of (x^2 + bx + a)/(bx^2 - x^3) 2. If x + 1/x = 99, find the value of 100x/(3x^2 + 103x + 3) Question from Tata Mc Graw-Hill NTSE b      Log On


   

Question 110861: 1. If x + a/x = b then find the value of (x^2 + bx + a)/(bx^2 - x^3)
2. If x + 1/x = 99, find the value of 100x/(3x^2 + 103x + 3)

Question from Tata Mc Graw-Hill NTSE book
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
x + a/x = b
(x^2 + bx + a)/(bx^2 - x^3)
(x^2+x(x + a/x)+a/(x^2(x + a/x )-x^3
(x^2+x^2 + a +a)/(x^3 + ax -x^3)
(2x^2+2a)/ax
.
x + 1/x = 99
x^2+1=99x eliminate fractions multiply each side by x
x^2-99x+1=0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-99x%2B1+=+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=%28-99%29%5E2-4%2A1%2A1=9797.

Discriminant d=9797 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--99%2B-sqrt%28+9797+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-99%29%2Bsqrt%28+9797+%29%29%2F2%5C1+=+98.9898979590785
x%5B2%5D+=+%28-%28-99%29-sqrt%28+9797+%29%29%2F2%5C1+=+0.0101020409215238

Quadratic expression 1x%5E2%2B-99x%2B1 can be factored:
1x%5E2%2B-99x%2B1+=+%28x-98.9898979590785%29%2A%28x-0.0101020409215238%29
Again, the answer is: 98.9898979590785, 0.0101020409215238. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-99%2Ax%2B1+%29

graph%28500%2C500%2C-10%2C110%2C-3000%2C1000%2Cx%5E2-99x%2B1%29
I'll let you plug the answers for x into the equation below to get the answers which should both be .25
100x/(3x^2 + 103x + 3)
.
Ed