I am looking to find the number of solutions to the following equation: $$x^3 +0.1=10x$$
Looking at the graph of the expressions on each side of the equation I understand that there are 3 solutions.
How would I go about illustrating this point algebraically? My 1st instinct is to equate everything to 0 $$x^3 –10x + 0.1=0$$ but I would have no idea how to factorize this equation. Any tips?
See: https://en.wikipedia.org/wiki/Cubic_equation#Depressed_cubic.
A depressed cubic in general form is $t^3 + pt + q$. The discriminant $\Delta = -(4p^3 + 27q^2)$. The given equation $x^3 - 10x + 0.1 = 0$ has a depressed cubic on the LHS and $\Delta = 3999.73 > 0$.
If ${\Delta >0,}$ the cubic has three distinct real roots. (See §Nature of the roots in the linked Wikipedia article.)
Since we want just the number of roots, there is no need to even factor or calculate the actual roots.