How to solve one equation for two unknowns

Question

I have an equation with two unknowns:

$$y = {25 \over (x-0.5)^2 } + {10 x}.$$

Is it possible to solve such an equation, I read somewhere that I can use differentiation but I'm not sure how.

Any help would be appreciated.

Answer 1

Well, you're trying to find:

$$\frac{\partial}{\partial x}\left\{\frac{\text{n}_1}{\left(x-\text{n}_2\right)^2}-\text{n}_3\cdot x\right\}=-\frac{2\cdot\text{n}_1}{\left(\text{n}_2-x\right)^3}-\text{n}_3=0\tag1$$

Now, we get one real solution:

$$x=\text{n}_2-\left(\frac{2\cdot\text{n}_1}{\text{n}_3}\right)^\frac{1}{3}\tag2$$

So:

$$\text{y}\left(\text{n}_2-\left(\frac{2\cdot\text{n}_1}{\text{n}_3}\right)^\frac{1}{3}\right)=\frac{3}{2^\frac{2}{3}}\cdot\text{n}_1^\frac{1}{3}\cdot\text{n}_3^\frac{2}{3}-\text{n}_2\cdot\text{n}_3\tag3$$