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1. ### Quadratic equation to find max and min

Do you mean the proof? You can prove it through Lagrange multipliers, or their is a trigonometrical proof too. For the Lagrange multipliers proof, your optimization function is ##acos(\theta)+bsin(\theta)## and restrain function is ##cos^2(\theta)+sin^2(\theta)=1##.
2. ### Quadratic equation to find max and min

Also, if you ever need, ## - \sqrt{a^2+b^2} \leq acos(\theta)+bsin(\theta) \leq \sqrt{a^2+b^2}##
3. ### Quadratic equation to find max and min

It is very simple. Let us consider ##2x^{2} +y^{2}=4## Then ##\frac{x^{2}}{2}+\frac{y^{2}}{4}=1## Here we make the substitution ##\frac{x^2}{2}=cos^2(\theta)## and ##\frac{y^2}{4}=sin^2(\theta)## And that is how we get it.
4. ### Quadratic equation to find max and min

##2x^2 + y^2 = 4## ##4x + y^2## I know you are done with the question, but I wanted to say that lagrange multipliers is not the best for this problem. Set ##x=\sqrt{2}cos(\theta), y=2sin(\theta)## So now we need to optimize ##4(\sqrt{2}cos(\theta)+sin^2(\theta) )##