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What is the smallest distance between the origin and a point on the graph of y = x^2 - 3?

 Jun 24, 2021
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The distance from a point on the graph to the origin can be expressed as, in terms of x, as \(\sqrt{x^2+(x^2-3)^2}=\sqrt{x^4-6x^2+9+x^2}=\sqrt{x^4-5x^2+9}\)

the quartic inside the radical is just a quadratic in terms of x^2, so we can rewrite it in vertex form:

\(\sqrt{x^4-5x^2+9}=\sqrt{x^4-5x^2+(\frac{5}{2})^2+9-(\frac{5}{2})^2}=\sqrt{(x^2-\frac{5}{2})^2+\frac{11}{4}}\)

since \((x^2-2.5)^2\)will always be nonnegative for real numbers, the minimum value for \(\sqrt{(x^2-2.5)^2+\frac{11}{4}}\) is just equal to \(\boxed{\frac{\sqrt{11}}{2}}\) (specifically when x is equal to \(\pm\sqrt{2.5}\))

P.S. my method seems like total overkill for this problem so if anyone has an easier solution, please post it

 Jun 25, 2021

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