4x^2 + y^2 = 25 and xy = 6.
I made y = 6/ x and got the equation :
4x^2 +( 6 / x )^2 = 25.
I thought i could then make it 4x ^ 2 + 36 / x^2 = 25.
Is that correct so far or completely wrong?
That's correct thus far. As long as $\displaystyle x \neq 0$ you can multiply through by $\displaystyle x^2$
$\displaystyle 4x^4 + 36 = 25x^2 \Leftrightarrow 4x^4-25x^2+36 = 0 \Leftrightarrow (x^2-4)(4x^2-9) = 0$
Both of those factors can be further factored using the difference of two squares to give x values.
Don't forget to check for extraneous solutions at the end if you use this method