# Why can a quadratic equation have only 2 roots?

It is commonly known that the quadratic equation $$ax^2+bx+c=0$$ has two solutions given by: $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ But how do I prove that another root couldn't exist?

I think derivation of quadratic formula is not enough....

• It's referred to as "The fundamental theorem of algebra" that counting multiple roots and complex roots, polynomials of degree n have n roots. You can google it. Dec 8, 2016 at 17:36
• "given the tag, I'm assuming we're working in R" and "Alternatively, you could note that the derivative has exactly one zero and thus by Rolle's theorem"..... Um...? Dec 8, 2016 at 17:41
• @MathematicsStudent1122 - pre-calculus students who have reached the point of quadratic equations and have heard that quadratics have two roots have likely encountered complex numbers. However, since they are pre-calculus, they are highly unlikely to be familiar with derivatives and Rolle's theorem. Dec 9, 2016 at 2:18
• @AtulMishra - Actually, derivation of the quadratic formula is enough: That derivation starts with an arbitrary solution to the polynomial, and shows that it has one of those two values. Therefore those two values are the only ones possible. Dec 9, 2016 at 2:20
• Can you assume that all polynomials can be written in the form $c(x-x_1)(x-x_2)\ldots$? Or do you need a proof of that too? Because if you can assume that, then the answer is trivially easy (just observer that having more than 2 roots implies a degree for $x$ greater than 2.) Dec 9, 2016 at 9:55

Suppose there are three distinct roots $x,y,z$. One has $$\begin{cases}ax^2+bx+c=0\\ay^2+by+c=0\\az^2+bz+c=0\end{cases}\Rightarrow\begin{cases}a(x^2-y^2)+b(x-y)=0\\a(x^2-z^2)+b(x-z)=0\end{cases}\Rightarrow\begin{cases}a(x+y)+b=0\\a(x+z)+b=0\end{cases}$$ It follows $$a(z-y)=0\Rightarrow z=y$$ which is a contradiction.

• your last system should be $a(x+y)+b = 0$ and $a(x+z)+b=0$. The rest stand, great proof. Dec 8, 2016 at 18:30
• I often mistaken for hurry (many times, because of my bad English, when I have finished my writing an equal answer has already been published.) Thank you. Dec 9, 2016 at 0:43
• @Wildcard Right. And it's also implicit that it's ok to divide by these terms because by the assumption of distinct roots neither of these can be zero. Dec 9, 2016 at 7:41
• It would be great if we had an $a \ne 0$ in there. Dec 9, 2016 at 15:09
• @DanielR.Collins - Recall the phrase "quadratic equation" in the title of the OP's posting. If $a=0$, one doesn't have a quadratic equation to start with. Thus, the condition $a\ne0$ is true automatically as long as one is working with (non-trivial!!) quadratic equations.
– Mico
Dec 11, 2016 at 9:21

I think derivation of quadratic formula is not enough....

Yes it is. The derivation is of the form if $ax^2+bx+c=0$, then $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$. The derivation is a proof if you pay attention.

The trickiest step is simply that if $y^2 = k$ for $k \geq 0$ then $y = \pm \sqrt k$, if you do not take this as evident.

• All that needed to be said. Dec 9, 2016 at 19:01
• @JackM thanks. I do think the answers regarding polynomial factorization over fields/integral domains, and how this relates to the fundamental theorem of algebra, are pretty useful. Dec 9, 2016 at 22:10

$$0 = ax^2 + bx + c$$

We solve this equation by completing the square. It offers up to two distinct solutions. The name we give to the general solutions is the quadratic formula. That's all there is.

If we consider the case of real solutions, and you think there may be a sneaky third solution, remember that $f(x) = ax^2 +bx +c$ can be plotted as below (depending on the sign of $a$). How many times could a parabola cross a horizontal line?

• this is but a specific degenerate case of 2D plane! Generalize it at very least to 3D complex space and you would easily see there are much more solutions !!! :-D Dec 9, 2016 at 12:00
• @BillDubuque Yes, that is all true. What's also true is that Math.SE exists for users of all levels. It's beneficial to provide responses that could meet varying expectations, unless the problem specifies. Dec 9, 2016 at 18:33
• This feels circular to me, don't you need to prove that all quadratics have a graph like that? Dec 9, 2016 at 19:31
• Useful <> Rigorous, it is possible for ideas that lack rigor to be useful, and moreover, useful to mathematical understanding. The point is, is it likely to be an accurate model? If so, its' potentially useful as an aid to understanding, regardless of whether or not we have a proof of its accuracy Dec 9, 2016 at 20:21
• @RemcoGerlich standupmaths made a "There is only One True Parabola" video demonstrating all parabolas are similar. Dec 9, 2016 at 21:23

A more general answer to this question lies in the following theorem:

Theorem If $P(x)$ is a polynomial of degree $n$, and $a$ is a value for which $P(a) = 0$, then $P(x) = (x - a)Q(x)$, where $Q(x)$ is a polynomial of degree $n - 1$.

This theorem is a simple consequence of polynomial long division. By long division, $P(x) = (x - a)Q(x) + R(x)$, for some polynomials $Q(x), R(x)$ with the degree of $R(x)$ less than the degree of $(x-a)$. But since $x - a$ is of degree 1, that means $R(x)$ is of degree $0$. I.e., $R(x) = R$, a constant.

But $P(a) = 0$, so $0 = (a - a)Q(a) + R$, and so $R = 0$ and we get just $P(x) = (x-a)Q(x)$. Since the degree of the product of two polynomials is the sum of their degrees, the degree of $P(x)$ is one greater than that of $Q(x)$, so the degree of $Q(x)$ must be $n-1$.

Now, if $P_n(x)$ is of degree $n > 0$ and $a_n$ is a root, then $$P_n(x) = (x - a_n)P_{n-1}(x)$$ for some $n-1$ degree polynomial $P_{n-1}(x)$. If $P_n(x)$ has another root $a \ne a_n$, then $a$ must also be a root of $P_{n-1}(x)$:

$$0 = P_n(a) = (a - a_n)P_{n-1}(a)$$ Since $a - a_n \ne 0$, we can divide it out to get $P_{n-1}(a) = 0$.

Conversely, if $a_{n-1}$ is a root of $P_{n-1}$, then $$P_n(a_{n-1}) = (a_{n-1} - a_n)P_{n-1}(a_{n-1}) = 0$$ So $a_{n-1}$ must also be a root of $P_n$ (which may be the same or different from $a_n$). We can also apply the theorem to $P_{n-1}$ and $a_{n-1}$: $$P_{n-1}(x) = (x - a_{n-1})P_{n-2}(x)$$ for some degree $n-2$ polynomial $P_{n-2}(x)$. By combining, we see that $$P_n(x) = (x - a_n)(x - a_{n-1})P_{n-2}(x)$$ As long as we can keep finding roots for the reduced polynomials, we can keep this up. If we can find $k$ such roots, $$P_n(x) = (x - a_n)(x - a_{n-1})(x - a_{n-2})...(x - a_{n+1-k})P_{n-k}(x)$$ Then $P_{n-k}(x)$ has to be a polynomial of degree $n-k$.

If we can find $n$ such roots, then $$P_n(x) = (x-a_n)(x-a_{n-1})...(x-a_1)P_0$$ where $P_0$ is a constant (a $0$-degree polynomial). $P_0 \ne 0$, since if it were we would have $P_n(x) = 0$ everywhere. But then the degree of $P_n$ would be $0$ (or less - some people define the degree of the $0$ to be $-\infty$), contrary to our original condition on $P_n(x)$. So in this case, $P_n(x)$ cannot have any other roots distinct from $a_1, a_2, ..., a_n$, since any other value would leave all factors in the expression non-zero.

So $P_n(x)$ can have at most $n$ roots.

The Fundamental Theorem of Algebra says that any non-constant polynomial over the complex numbers has a root. This theorem requires a substantial development of the properties of complex numbers to prove. But by it, we see that the process above does not terminate until you get to the constant. Thus a polynomial of degree $n$ will always have exactly $n$ roots $a_1, a_2, ..., a_n$. But remember that the $a_i$ values do not have to be distinct. The number of times a particular value occurs in this list is called the multiplicity of the root. So you only get $n$ if you count the roots by their multiplicity.

Hint  If $$f(x)\,$$ is a polynomial of $$\color{#0a0}{{\rm degree}\,2}\,$$ with coef's in a $$\rm\color{#c00}{field}$$ (or $$\rm\color{#c00}{domain}$$) $$F$$ (e.g. $$\,\Bbb Q,\Bbb R,\Bbb C,\Bbb Z_p)$$ and $$\,f\,$$ has $$\,2\,$$ distinct roots $$\,a\neq b,\,$$ then applying the Bifactor Theorem below we deduce that $$\,f(x) = c(x\!-\!a)(x\!-\!b)\,$$ for $$\,\color{#0a0}{0\neq c}\in F.\,$$ Thus if $$\,d\neq a,b\,$$ then $$\,f(d) = c(d\!-\!a)(d\!-\!b)\ne 0\,$$ since each factor is $$\ne 0\,$$ (recall $$\,x,y\ne 0\,\Rightarrow\,xy\ne 0\,$$ in a $$\rm\color{#c00}{field}$$). Thus a $$\rm\color{#0a0}{quadratic}$$ has at most $$\,\color{#0a0}2\,$$ roots.

Bifactor Theorem $$\$$ Suppose that $$\rm\,a,b\,$$ are elements of a field $$\rm\,F\,$$ and $$\rm\:f\in F[x],\,$$ i.e. $$\rm\,f\,$$ is a polynomial with coefficients in $$\rm\,F.\,$$ If $$\rm\ \color{#C00}{a\ne b}\$$ are elements of $$\rm\,F\,$$ then

$$\rm f(a) = 0 = f(b)\ \iff\ f\, =\, (x\!-\!a)(x\!-\!b)\ h\ \ for\ \ some\ \ h\in F[x]$$

Proof $$\,\ (\Leftarrow)\,$$ clear. $$\ (\Rightarrow)\$$ Applying Factor Theorem twice, while canceling $$\rm\: \color{#C00}{a\!-\!b\ne 0},$$

$$\begin{eqnarray}\rm\:f(b)= 0 &\ \Rightarrow\ &\rm f(x)\, =\, (x\!-\!b)\,g(x)\ \ for\ \ some\ \ g\in F[x]\\ \rm f(a) = (\color{#C00}{a\!-\!b})\,g(a) = 0 &\Rightarrow&\rm g(a)\, =\, 0\,\ \Rightarrow\ g(x) \,=\, (x\!-\!a)\,h(x)\ \ for\ \ some\ \ h\in F[x]\\ &\Rightarrow&\rm f(x)\, =\, (x\!-\!b)\,g(x) \,=\, (x\!-\!b)(x\!-\!a)\,h(x)\end{eqnarray}$$

Remark  More generally, by inductively iterating the Factor Theorem (as we did above) we deduce that a nonzero polynomial $$\,f\,$$ over a field (or domain) has no more roots than its degree $$\,n.\,$$ Indeed if $$\,f\,$$ has $$\,\ge n\,$$ distinct roots $$\,r_i$$ then inductively applying the Factor Theorem shows that $$\,f = c(x\!-\!r_1)\cdots (x\!-\!r_n),\,$$ so $$\ r\ne r_i\Rightarrow\, f(r)= c(r\!-\!r_1)\cdots (r\!-\!r_n) \ne 0\,$$ by all factors are $$\ne 0.\,$$ Thus $$\,f\,$$ has at most $$\,n\,$$ roots.

The above root-bound property characterizes (integral) domains (commutative rings $$\ne \{0\}$$ which satisfy $$\rm\,ab=0\,\Rightarrow\, a=0\,$$ or $$\rm\,b=0),\,$$ viz. a ring $$\rm\: D\:$$ is a domain $$\iff$$ every nonzero polynomial $$\rm\ f(x)\in D[x]\$$ has at most $$\rm\ deg\ f\$$ roots in $$\rm\:D.\:$$ For a simple proof see this answer, where I illustrate it constructively in $$\rm\: \mathbb Z/m\:$$ by showing that, given any $$\rm\:f(x)\:$$ with more roots than its degree, we can quickly compute a nontrivial factor of $$\rm\:m\:$$ via a quick $$\rm\:gcd.\,$$

The quadratic case of this result is at the heart of some integer factorization algorithms, which e.g. attempt to factor $$\rm\:m\:$$ by searching for a square-root of $$1$$ that is nontrivial $$(\not\equiv \pm1)$$ in $$\rm\: \mathbb Z/m.$$

Beware that there are very simple examples of failure in non-domains, e.g. if $$\,ab=0, a,b\neq 0\,$$ then $$\,ax\,$$ has at least $$2$$ roots $$\,b,0,\,$$ and $$\,(x-a)(x-b)\,$$ has at least $$\,4\,$$ roots $$\,a,b,0,a+b\,$$ if $$\,a\neq b.\,$$ A simple concrete case is in $$\,\Bbb Z_8 =$$ integers $$\!\bmod 8\!:\,$$ $$\rm{odd}^2= 1\,$$ so $$\,x^2-1\,$$ has $$\,4\,$$ roots $$\,\pm1,\pm 3.$$

• See this answer for a generalization from domains to any commutative ring. Jul 14, 2017 at 14:14

Suppose it has three roots and $a\neq0$.

The hypotheses of Rolles Theorem are satisfied then there will exist two roots of the derivative and a root of the second derivative which is a constant $(=2a)$.

Let $$r=\frac{-b+\sqrt{b^2-4ac}}{2a},\quad s=\frac{-b-\sqrt{b^2-4ac}}{2a}.$$ Simple calculation shows that $$r+s=-\frac ba\quad\text{ and }\quad rs=\frac{b^2-(b^2-4ac)}{4a^2}=\frac ca.$$ Thus $$a(x-r)(x-s)=a[x^2-(r+s)x+rs]=ax^2+bx+c.$$ If $t$ is any root of the quadratic equation $ax^2+bx+c=0,$ then $$a(t-r)(t-s)=at^2+bt+c=0.$$ Since $a\ne0$ this means that $$(t-r)(t-s)=0$$ whence $$t-r=0\quad\text{ or }\quad t-s=0,$$ i.e., $$t=r\quad\text{ or }\quad t=s.$$

I think I can simplify the 'polynomial long division' answer. The special case of polynomial long division says that, for any polynomial $P$, and any real number $a$, $$P(x) = Q(x)(x-a) + R$$ for some polynomial $Q$ and constant $R$ (use long division to divide $P$ by $x-a$ and observer that $R$ is required to be a constant since it must have lower degree than $x-a$, which is a first-degree polynomial).

The above is true for all $x$, so substituting $x=a$ we get $$P(a) = Q(a)(a-a) + R$$ Obviously, $a-a=0$, so $R=P(a)$.

If $a$ is a 0 of $P$ ($P(a)=0$), then $R=0$, so $x-a$ divides $P(x)$.

Now, if we have any distinct root $a$ of a quadratic polynomial $P$, we know $$P(x) = Q(x)(x-a)$$ $Q$ must be a first-degree polynomial, since anything higher-degree, multiplied by a first-degree polynomial, would produce a higher-than-second-degree polynomial. So $$P(x) = (x-b)(x-a)$$ Now, assuming we're working over an integral domain (which $\mathbb R$ and $\mathbb C$ both are), $$P(x) = 0 \Rightarrow x-b=0 \text{ or } x-a=0$$ So $a$ and $b$ are the only zeros of $P$ (although it is possible that $a=b$).

Suppose not. Then there are at least 3 roots and so $$P(x)=(x-x_1)(x-x_2)(x-x_3)Q(x)$$ This is at least a cubic so this cannot happen.

If you wrote the theorem out, it would look like this

THEOREM: Let $a,b,$ and $c$ be real numbers with $a \ne 0$. Then $ax^2+bx+c=0$ if and only if $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

That means

if $ax^2+bx+c=0$, then $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

and

if $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, then $ax^2+bx+c=0$.

Suppose $$a \neq 0$$. Then $$ax^2+bx+c=0$$ has the same set of solutions as $$x^2+\frac{b}{a}x+\frac{c}{a}=0$$ which means that we can reduce all equations to

$$x^2+px+q=0$$

without losing generality.

Suppose $$q=0$$, then it is $$x^2+px=0$$ when either $$x=0$$ or $$x+p=0$$, but $$x+p=0$$ has only one solution by definition $$x=-p$$

Next, take then $$q \neq 0$$, when

$$x(x+p)=-q$$

Take $$y=x+\frac{p}{2}$$ which makes

$$(y+\frac{p}{2})(y-\frac{p}{2})=y^2-\frac{p^2}{4}=-q$$

or

$$y^2=-q+\frac{p^2}{4}=b$$

So it comes down to a question of how many different roots square root may have.

Write two of them which we suspect:

$$y_1^2=b$$

$$y_2^2=b$$

$$y_1^2-y_2^2=(y_1-y_2)(y_1+y_2)=0$$

So it is either $$y_1=y_2$$ which gives the same solution or $$y_1=-y_2$$ which connects the solutions in such a way that there cannot be two different negative values of the same number. Since the connections of reduced solution $$y_1,y_2$$ are connected with $$y_1=-y_2$$ the starting set of solutions must have the same cardinality: no more than $$2$$.

Finally if $$a=0$$ there can be only one solution by definition.