Discriminant formula issue Could you please explain to me, how to get the formula of discriminant ? How can I visualize it, any articles, lectures? 
I can memorize it $b^2 - 4ac$ But, want to understand it.
Thanks. 

Consider the polynomial
$$ax^2+bx+c$$
First I would like to make this polynomial a multiple of a polynomial with its  first term a perfect square and the second term even. To do this, we multiply by $\frac{4a}{4a}.$
$$\begin{align} \frac{1}{4a}(4a^2x^2 + 4abx+4ac) &= \frac{1}{4a}((2ax)^2+2b(2ax)+4ac) \\
& = \frac{1}{4a}((2ax)^2+2b(2ax)+b^2-b^2+4ac)\\
& = \frac{1}{4a}(\bigg((2ax)^2+2b(2ax)+b^2\bigg)-(b^2-4ac))\\
& = \frac{1}{4a}((2ax+b)^2-(b^2-4ac))\\
& = \frac{1}{4a}((2ax+b)^2-\sqrt{b^2-4ac}^2)\\
& = \frac{1}{4a}(2ax+b+\sqrt{b^2-4ac})(2ax+b-\sqrt{b^2-4ac})\\
 \end{align}$$
You can easily see from this that when the discriminant is zero, the second and third factors both have the same root. When the discriminant is positive, the roots are different. And when the discriminant is negative, there are no real roots.
