Find rectangular equation from parametric equation???

Question

Find rectangular equation from parametric

$ x = t^{2} + t $

$ y = t^{2} - t $

I tried finding the equation but I am stuck here:

$ x - t^{2} = t $

$ y = t^{2} - t $

$ y = t^{2} - ({x - t^{2}}) $

$ y = t^2 - x + t^2 $

$y = 2t^2 - x $

Is there even a parametric equation for this?

Answer 1

Hint: $x-y=2t$, hence $t=(x-y)/2$. Substituting in the first equation gives $$x=\frac{1}{4}(x-y)^2+\frac12(x-y)\iff4x=x^2-2xy+y^2+2x-2y$$ $$\iff x^2-2xy+y^2-2x-2y=0.$$

Answer 2

$y=x-2t\\ 0=t^2+t-x\\ -2t = 1\pm\sqrt{1+4x}\\ y=1+x+\sqrt{1+4x}\\ y=1+x-\sqrt{1+4x}$

Answer 3

We have $$x+y=2t^2\iff t^2=\frac{x+y}2, x-y=2t\iff t=\frac{x-y}2$$

$$\implies\left(\frac{x-y}2\right)^2=\frac{x+y}2\iff (x-y)^2=2(x+y)$$