I'm doing a problem in which I have to find the minimum distance between the origin and a point on a parabola. I understand that the minimum of a function and the minimum of the square of the function are the same. The distance function when considering that one point is the origin should be $d =\sqrt{x^2 + y^2}$, making the squared function $d = x^2 + y^2$. However, when solving for $y$ so that I can make the equation solely in terms of $x$, I get $y = \sqrt{-x^2}$, and substituting that for $y$ in the original equation would negate that square root and result in $x^2 -x^2$, or $0$. Hopefully I just overlooked something, but that obviously seems wrong if I'm trying to find a distance function I can derive and get a minimum.
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$\begingroup$The general point on the parabola $y=x^2$ has two coordinates: $(t,t^2)$ for $t$ a real. Use this in the distance squared to $(0,0)$ formula. Multiply out, find derivative, ETC.
I don't think parabola formula is $y=x^2$ since then min distance clearly zero (no work to do). But same idea works if $y=ax^2+bx+c$ where $a \neq 0.$ then general point is $(t,at^2+bt+c)$ and proceed as above.
Note: It just struck me that your problem may have been in using $x,y$ for both the point on the parabola, and in applying $x^2+y^2$ for distance to origin. If you keep $x,y$ for the parabola point, then must solve for one of $x,y$ in terms of the other to set things up. That's why a good idea is to use another variable like $t$ to get the parabola parametrized in that new letter, like $(t,t^2+1)$ or what have you. Initially check if $(0,0)$ already on the parabola [if so stop, min distance $0.$] Otherwise in $d^2=x^2+y^2$ you have to substitute the coordinates into that, e.g. in my made-up example $t^2+(t^2+1)^2.$ Good luck.
I looked at your example $x^2-2x+5$ and found the same numeric value for $x$ as you mentioned in comments. Then I applied the cubic formula and got that, if $u=\frac{-1+\sqrt{55}}{4},$ then$$x=u^{1/3}+\frac{3}{2u^{1/3}}+1.$$A google search should get you to several versions of the cubic formula.
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