We know that when a quadratic equation $ ax^2 + bx + c = 0 $ has zero discriminant value, then the quadratic equation has only "one root". But why do some mathematician call it "double root" and also that the equation still has two roots, but their roots are double. When we draw the equation in the coordinate plane, we can see that the graph of the equation will just touch the $ x $-axis at only one point.
The second question is: We know that the sum of the roots of a quadratic equation $ ax^2 + bx + c = 0 $ is $ -\dfrac{b}{a} $. But when the discriminant of that equation is zero, the sum is also $ -\dfrac{b}{a} $. It doesn't make sense. For example, if we have quadratic equation $ x^2 + 4x + 4 = 0 $, we can factor it as $ \left(x + 2\right)^2 = 0 $ and so $ x = -2 $. Thus, the sum is just $ -2 $. But with "sum of the roots" formula, the sum is $ -4 $. Which one is correct? Is this the reason we call it double root?
Thanks in advance :)
Consider the assertion “the polynomial $(x-a)(x-b)$ has two roots”. Is it true? Well, yes if $a\ne b$, but no if $a=b$. If $a=b$, then that polynomial is $(x-a)^2$ and then we say that $a$ is a double root. Then if we count the roots with their multiplicity, (a double root is counted twice, a triple root is counted three times, …), a polynomial with degree $n$ will always have $n$ roots (at least over the complex numbers).
And, going back to quadratics, if we say that the roots of $(x-a)^2$ are $a$ and $a$, then $a+a=2a$ and $(x-a)^2=x^2-2ax+a^2$ and indeed the sum of the roots is minus the coefficient of $x$.