Is this the correct strategy to convert this parametric to rectangular? Do I need to further simplify the answer or is this good enough?
Hello, CrazyLond!
Yet another approach . . .
$\displaystyle \begin{array}{cccc}x &=&t^2+t & [1] \\ y &=&t^2-t & [2] \end{array}$
Subtract [1] - [2]: .$\displaystyle x-y \:=\:2t\quad\Rightarrow\quad y \:=\:\frac{x-y}{2}$
Substitute into [2]: .$\displaystyle y \:=\:\left(\frac{x-y}{2}\right)^2 - \left(\frac{x-y}{2}\right) $
This simplifies to: .$\displaystyle x^2 - 2xy + y^2 - 2x - 2y \:=\:0$
I believe this is the parabola: $\displaystyle y^2 = \sqrt{2}\,x$ rotated 45° CCW.
. . (But don't quote me!)