Is there exist any real solution of the equation...

Question

While solving a question given by my friend I found a problem in finding the real roots, I tried the question many times but I could not find any real solution of that equation.

The equation was:

$5x^2-5y^2-8xy-2x-4y+5=0$

Please help me to find its real solution

Answer 1

What you have looks like:$$5x^2-5y^2-8xy-2x-4y+5=0$$

$$ 5^2+x(-8y-2)-5y^2-5y-5=0$$

$$ 5x^2-2x(4y+1)-y(5y+4)$$

See that it is a quadratic equation in terms of $x$. Solve for $x$ in terms of $y$.

Side Note: It represent a hyperbola and integer solutions are $(x,y)$=$(-5,10),(-1,2)$.

SEE Wolfram Alpha.

Answer 2

On rewritting the equation as

$-5y^2-y(8x+4)+5x^2-2x+5=0$

and solving it using quadratic formula we get

$y=\frac{(8x+4) \pm 2\sqrt{(-9x^2+26x-21}}{10}$

try to solve this for $y$ in terms of $x$