SOLUTION: The denominator of a fraction is an integer that is 4 more than the square of the numerator. if the fraction is reduced to lowest terms, it equals 3/20. find the numerator

Algebra ->  Rational-functions -> SOLUTION: The denominator of a fraction is an integer that is 4 more than the square of the numerator. if the fraction is reduced to lowest terms, it equals 3/20. find the numerator      Log On


   

Question 230130: The denominator of a fraction is an integer that is 4 more than the square of the numerator. if the fraction is reduced to lowest terms, it equals 3/20. find the numerator
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x%2F%284%2Bx%5E2%29+=+3%2F20
To get rid of the fractions we will multiply by the Lowest Common Denominator (LCD). The LCD here is: 20(4+x^2).
20%284%2Bx%5E2%29%28x%2F%284%2Bx%5E2%29%29+=+20%284%2Bx%5E2%29%283%2F20%29
Cancel common factors on each side:

leaving:
20%2Ax+=+%284%2Bx%5E2%29%2A3
or
20x+=+12+%2B+3x%5E2
Since this is a quadratic equation, we will get one side equal to zero:
0+=+3x%5E2+-+20x+%2B+12
... then factor it (or use the quadratic formula):
0+=+%283x+-+2%29%28x-6%29
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
3x-2+=+0 or x-6+=+0
Solving these we get:
3x+=+2 or x+=+6
x+=+2%2F3 or x+=+6

Since the problem specifies integers, we must reject x = 2/3. So the only answer to the problem is x=6.