### Instruction: Find the minimum value of
\[\frac{x^2 + 7}{\sqrt{x^2 + 3}}\]over all real numbers $x.$.### Response: To minimize a fraction, I want to make the numerator as small as possible and the denominator as large as possible.
The numerator is a quadratic function that has a minimum value of 7, which occurs when x = 0.
The denominator is a square root function that is always positive and has a minimum value of $\sqrt{3}$, which occurs when x = 0.
So the minimum value of the fraction is $\frac{7}{\sqrt{3}}$, which occurs when x = 0.
I can simplify this by rationalizing the denominator and multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$, which gives $\frac{7\sqrt{3}}{3}$.