Given that 81h/2e + 2e/h = 18,
Find the value of e : h
Does anyone here know how I should go about doing this? I confused myself mid-way when attempting to solve the equation >n<;
Assuming this is $\displaystyle \displaystyle \begin{align*} \frac{81h}{2e} + \frac{2e}{h} = 18 \end{align*}$, then
$\displaystyle \displaystyle \begin{align*} h \left(\frac{81h}{2e} + \frac{2e}{h} \right) &= 18h \\ \frac{81}{2e}h^2 + 2e &= 18h \\ \frac{81}{2e}h^2 - 18h + 2e &= 0 \end{align*}$
And now solve for h using the Quadratic Formula.
Instead of solving $\displaystyle \frac{81}{2e}h^2 - 18h + 2e = 0$ for h and then dividing e by that h, you could divide both sides by e to get $\displaystyle \frac{81}{2}\left(\frac{h}{e}\right)^2 - 18\frac{h}{e} + 2 = 0$ and solve it to find $\displaystyle \frac{h}{e}$. This can also be done starting from the initial equation $\displaystyle \frac{81h}{2e} + \frac{2e}{h} = 18$. If we denote e / h by x, then the equation becomes $\displaystyle \frac{81}{2}\frac{1}{x} + 2x = 18$, which turns into a quadratic equation in x after multiplying by x.
When I was in a freshman in high school, I took part in a math competition designed for sophomores. There was a quadratic equation, and I could not solve it because we had not covered the quadratic formula yet though I believe we had covered completing the square. Later my math teacher told me that I could have derived the quadratic formula by completing the square. So don't be afraid to study the quadratic formula and how to derive it. I personally never use factorization but instead apply the quadratic formula.