I would like to know how to solve the following problem:
Find the range of values of the parameter $m$ for which the equation $2x^2 – mx + m = 0$ has no real solutions.
I know I have to use the quadratic formula and the response is $0 < m < 8$.
But what I don't know is how to proceed to find this answer. Thanks for your help.
Best Answer
No, you don't have to use the quadratic formula. Since\begin{align}2x^2-mx+m&=2\left(x-\frac m4\right)^2+m-\frac{m^2}8\\&=2\left(x-\frac m4\right)^2+\frac{8m-m^2}8\end{align}it s clear that your equation has no roots if and only if $8m-m^2>0$. And, since $8m-m^2=m(8-m)$, this occurs if and only if $m\in(0,8)$.