Real and distinct roots of a cubic equation The real values of $a$ for which the equation $x^3-3x+a=0$ has three real and distinct roots is
The derivative cancels at $x=1$ and $x=-1$. To these correspond a maximum value of $a+2$ and a minimum value of $a-2$. In order to have three real roots, you need three $x$ intercepts; this means that you must have $a+2&gt;0$ and $a-2&lt;0$. So, the condition is $|a|&lt;2$.
Here is a graphical approach. Note that '+ a' shifts the graph up or down.

1) Disregard '+ a'.

2) Graph $ f(x) = x^3 - 3x $

http://www.wolframalpha.com/input/?i=plot%28x%5E3+-+3x%29#

3) from the graph we see that we must find the y values at the local extrema. You did that already getting y = 2 and -2

4) It should be obvious now by inspection that a must be between -2 and 2

Final answer ,

-2 &lt; a &lt; 2