How to convert this parametric equation into a Cartesian equation?

Question

Sketch the curve by using the parametric equation to plot points. Indicate with arrow the direction in which the curve is traced as $t$ increases.

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

I tried to convert this parametric equation into a Cartesian equation, but I didn't know how. Please help.

Answer 1

HINT:

On addition we have $\displaystyle 2t^2=x+y\iff t^2=\frac{x+y}2$

On subtraction, $\displaystyle 2t=x-y\iff t=\frac{x-y}2$

Can you eliminate $t$ from here?

Answer 2

HINT : $t^2=x-t=y+t$. So, you can represent $t$ by $x,y$.

Since we have $t=\frac{x-y}{2}$, we have $$x=\left(\frac{x-y}{2}\right)^2+\frac{x-y}{2}.$$

We can get the following equation with $x$ and $y$ :

$$x^2-2xy+y^2-2x-2y=0.$$