x = y/(y^2 - 16), solve for y

Question

solve equation for y

Answer 1

x=y/(y²-16) ⇔ x(y²-16)=y ⇔ y²-y/x-16=0. Solve quadratic equation with respect to y.

 

y_1,2=[1/x±√(1/x²+64)]/2=[1±√(1+64x²)]/(2x)

 

 

 

 

Answer 2

Sometimes the answer is not well defined, like in this case. This problem may be solved by the quadratic method, like Kirill did, or by completing the square for y² - y/x, by adding and subtracting 1/4x². The problem then reduces to

 

(y² - y/x + 1/4x²) -16 - 1/4x² = 0, or

(y - 1/2x)² = 16 + 1/4x² .

Then y - 1/2x = ± √(16 + 1/4x²) , and

y = 1/2x ± √(16 + 1/4x²), which reduces to the same answer Kirill got. 

 

This method does not require memory of the quadratic formula, but that formula provides a fool proof method of solving quadratic equations.