Calculate the number of solutions Calculate the number of solutions presented by the equation.
$$\sqrt{1-x}+\sqrt{1-2x}+\sqrt{1-4x}=x^2+2$$
What I thought: The LHS is concave, the RHS is convex

Hint : If $X= \sqrt{a}+ \sqrt{b}+\sqrt{c}$ ... keep squaring and rearranging ...
\begin{eqnarray*}
X &=& \sqrt{a}+ \sqrt{b}+\sqrt{c} \\
\frac{X^2-a-b-c}{2} &=& \sqrt{ab}+ \sqrt{bc}+\sqrt{ca} \\
\left(\frac{X^2-a-b-c}{2} \right)^2 &=& 2\sqrt{abc}(\sqrt{a}+ \sqrt{b}+\sqrt{c})=2\sqrt{abc}X. \\
\end{eqnarray*}
Now square one final time & what order is the polynomial ?
