If a quadratic equation can have real and equal roots, then why don't we say it has one root?

Question

Suppose $ax^2+bx+c$ is a quadratic equation with $D=0$

So it has the roots $x=\frac{-b}{2a},\frac{-b}{2a}$ which are real and equal

Why don't we just say it has one root which would be $x=\frac{-b}{2a}$?

Answer 1

Information is useful, and we do not like to throw it away carelessly. As it turns out, the correct count of the multiplicities of the roots is important information for many mathematical applications of polynomials.

For a quadratic equation $a x^2 + bx + c = 0$ with discriminant $D = b^2-4ac$, if $D=0$ then the left hand side factors as $$a x^2 + bx + c = a(x-r)^2 $$ where $r=-\frac{b}{2a}$, and in this case we say that $r$ is a double root or a root of multiplicity $2$.

Here is an application: The question of whether a root $r$ is single or double is connected with the graph of the equation $y=ax^2 + bx + c$. That graph is a parabola, and it has reflective symmetry across the vertical line passing through the vertex of the parabola. As it turns out, if $r$ is a root of the equation then $r$ is a double root if and only if the point $(r,0)$ is the vertex of the parabola, which occurs if and only if the parabola is tangent to $x$-axis at the point $(r,0)$.

When you study polynomials of higher degree, you will learn that for any root $r$, the graph of the polynomial is tangent to the $x$-axis at the point $(r,0)$ if and only if $r$ of multiplicity $d\ge 2$ (meaning that $d$ is the highest exponent for which $(x-r)^d$ is a factor of the polynomial).

And there is still more information to be obtained and applied. For example, if $r$ is a root of multiplicity $3$ of some polynomial, then the point $(r,0)$ on the graph of that polynomial is tangent to the $x$-axis and that point is an inflection point of the graph.

Think, for example think about the graph of $y=x^3 - 3x^2 - 3 + 1$, which factors as $y = (x-1)^3$. Its graph is tangent to the $x$-axis at the point $(1,0)$ and that point is an inflection point.

Answer 2

If a quadratic equation can have real and equal roots, then why don't we say it has one root?

Sometimes we do, for example when saying that the equation $x^2 - 2x + 1 = 0$ has a unique real root $\,x=1\,$.

Sometimes we do without saying it explicitly, for example the RMS-AM inequality $\,\dfrac{x^2+1}{2} \ge \left(\dfrac{x+1}{2}\right)^2\,$ is said to have one equality case when $x=1$, though that's a double root of the associated quadratic.

Other times, depending on the context and emphasis, it may be important to specify the multiplicity. A double root is not only a root, but also a critical point of the quadratic. Referring to it as just "one root" overlooks and discards the significance of it being in fact a double root.

Yet other times, we may use different words to disambiguate, for example when saying that the equation $x^2 - 2x + 1 = 0$ has a unique real solution, which is a double root of the quadratic.

There is also a potential for confusion related to the FTA, usually remembered as "an $n^{th}$ degree complex polynomial has $n$ complex roots, counting multiplicities". In some algebra related contexts, this gives "root" an implicit meaning of "counting multiplicities". In such contexts, saying that a quadratic has one root could be seen as ambiguous at best, or plain wrong at worst.