Determining whether a quadratic has a maximum or minimum

So I've learnt that quadratics equations with a positive coefficient on the squared term have a minimum and a maximum if the coefficient is negative. But if we rearrange the quadratic and change the signs of the squared term, doesn't that mean the equation's maximum would change to a minimum and vice versa? e.g $$x^2 + 5x + 10 = 0$$ becomes $$-x^2 - 5x - 10 =0$$ even though we've not changed the equation, it's now got a maximum rather than a minimum. What am I misunderstanding here?

• You're perfectly right. The equation doesn't change, but the inequation is reversed. – Bernard Mar 14 '19 at 23:55
• It's functions that have minimums and maximums. Not equations. – Michael Rybkin Mar 14 '19 at 23:56
• You haven't (really) changed the equation, but you have changed the quadratic. The question of a maximum or minimum refers to something like $x^2+5x+10$, not something like $x^2+5x+10=0$. – David Mar 14 '19 at 23:57
• The $x$ values where $x^2+ax+b=0$ are the same as those where $-x^2-ax-b=0,$ but $x^2+ax+b$ differs from $-x^2-ax-b$ at other values of $x$ – J. W. Tanner Mar 15 '19 at 0:11

If $$f(x)=0$$, the maximum and minimum of $$f(x)$$ are both $$0$$ (unless there are no $$x$$ such that $$f(x)=0$$).