Question

The difference of a positive integer and its inverse is 15/4. Find the integer.

Answer 1

X - 1/X = 15/4
X/1 - 1/x = 15/4
X • (X/1) - 1/X = 15/4
X ^2 - 1/X = 15/4
X • (X ^2 - 1/X) = 15/4 • X

X^2 - 1 = 15x/4
4 • (X ^2 - 1) = 15x/4 • 4
4(X^2 - 1) = 15x
4x ^2 - 4 = 15x
4x^2 - 15x - 4 = 0
4x^2 - 15x = 4
X(4x - 15) = 4.
X = 4

4x - 15 = 4
4x = 19
4x/4 = 19/4
X = 19/4.

I believe the solution is 4.

4/1 - 1/4 = 15/4
16/4 - 1/4 = 15/4
15/4 = 15/4.

Answer 2

Answer: The positive integer is 4.

Explanation: If we have the integer x, we need to solve:

x - 1/x = 15/4

we multiply all the equation by x.

x^2 - 1 = (15/4)*x

x^2 - (15/4)*x - 1 = 0

now we can solve this quadratic equation and find the value of x

the solutions are:

`x = \frac{15/4 +/-\sqrt{(15/4)^(2) - 4*-1*1 } }{2} = \frac{15/4 +/- \sqrt{(15/4)^(2) + 4 }  }{2}  = (3.75 +/- 4.25)/(2)`

So the solutions are:

x = (3.75 - 4.25)/2 = -0.25

this is not integer nor positive, so this is not the solution we are looking for.

and the other solution is:

x = (3.75 + 4.25)/2 = 4

this is integer and positive, so this is the solution we are looking for.