If you don't remember the quadratic formula and don't want to look it up, the graphical intuition is super simple, if you're comfortable connecting the graphica

If you don't remember the quadratic formula and don't want to look it up, the graphical intuition is super simple, if you're comfortable connecting the graphical to the algebraic bits. A quadratic equation is the same as finding where a parabola crosses zero. The formula for that is messy unless the parabola is centered around x=0, in which case it's easy. Then it's just a "standard" parabola (y=x^2), stretched horizontally and/or moved up and down. The equation for that shape is 0=ax^2-k, so x^2=k/a, which is super simple. So then you just have to figure out how to turn your original problem (a parabola that's not centered at x=0) into the easy one (a parabola that is centered at x=0). The answer is to just slide your axes over until it's centered. That is, define a coordinate z = x-c such that your parabola is centered at z=0. To do that, just plug in z+c=x into the original equation, and solve for a c that makes the middle term disappear (you want to end up with y=az^2 + 0z + b, per my second paragraph). Now you have the problem that's easy to solve. And there's the added benefit that whatever problem this comes up in is likely easier to think about in terms of this z than in x. This method is just some trivial steps, which makes it really easy to remember. It's easy to slide a shape left and right, and it's easy to solve x^2=k.