How can one isolate x in a formula of the form:$ (x-20)^{2} = -(y-40)^{2} - 525$?

Question

I am trying to isolate x in the equation $$(x-20)^{2} = -(y-40)^{2} - 525.$$ How can I do it?

Answer 1

If you are looking for real solutions, there can be none. The left side is nonnegative, and the right side is strictly negative.

Otherwise, for complex solutions, you can take square roots of both sides and add $20$. Don't forget that you get two equations (with $\pm$) when you take square roots:

$$x = 20 \pm i\sqrt{(y-40)^2 + 525}$$

This is shorthand notation for the two equations

$$x = 20 + i\sqrt{(y-40)^2 + 525}$$ $$\textrm{or}$$ $$x = 20 - i\sqrt{(y-40)^2 + 525}$$

Answer 2

If $x$ and $y$ are real, the right side is negative while the left side is non-negative, so the equation never holds.

Answer 3

$$(x-20)^{2} = -(y-40)^{2} - 525$$

Square root both sides:

$$x-20= \pm \sqrt{-(y-40)^{2} - 525}$$

Rearrange:

$$x= 20 \pm \sqrt{-(y-40)^{2} - 525}$$

Or equivalently,

$$x= 20 \pm i\sqrt{(y-40)^{2} + 525}$$

Answer 4

$x=(\sqrt {(y-40)^ 2+525})i+20$ where $i=\sqrt {-1}$.

If domain of x is real number set than there is no solution.