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If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots?

I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and rational numbers are of the form $\frac pq$.

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This is incorrect, as roots are real if $b^2\geq 4ac$ Maybe you mean rational? –  Ishan Banerjee Mar 24 '13 at 14:10
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You are like saying: I give you this problem. DO IT! –  ᴊ ᴀ s ᴏ ɴ Mar 24 '13 at 14:12
    
Why is this now gaining downvotes? It was +7/0 and now +7/-3 –  Mr.ØØ7 Apr 9 '13 at 14:54
    
@exploringnet: if this is revenge downvoting, the scripts will have to deal with it. Otherwise, there is nothing much to be done. –  robjohn Apr 9 '13 at 18:32
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9 Answers 9

up vote 9 down vote accepted

Consider a quadratic equation of the form $a\cdot x^2 + b\cdot x + c = 0$. The only way, it can have rational roots IFF there exist two integers $\alpha$ and $\beta$ such that

$$\alpha \cdot \beta = a\cdot c\tag1$$ $$\alpha + \beta = b\tag2$$ $$ Explanation\left\{ \begin{align} if\,\alpha\cdot \beta &= a\cdot c,\\ \frac{\alpha}{a} &= \frac{c}{\beta}\\ a\cdot x^2 + b\cdot x + c& = a\cdot x^2 + (\alpha + \beta)\cdot x + c\\ & = a\cdot x^2 + \alpha\cdot x + \beta\cdot x + c\\ & =a\cdot x(x + \frac{\alpha}{a}) + \beta\cdot (x + \frac{c}{\beta}))\\ & =a\cdot x(x + \frac{\alpha}{a}) + \beta\cdot (x + \frac{\alpha}{a}))\\ & =(x+\frac{\alpha}{a})\cdot (a\cdot x + \beta)\\ &\text {As a Quadratic equation has only two roots,}\\ &\text {there would be no other way to factorize the equation} \end{align}\right. $$ $$\text{Reason }\alpha,\beta\in\mathbb{Z}\begin{cases} \text{Given a Ring R, with two operations }\left\{⋅,+\right\},\text{ on }\mathbb{Q}\\ \text{and if } \alpha,\beta \in \mathbb{Q},\alpha\cdot \beta \in \mathbb{Z},\alpha+\beta\in\mathbb{Z}\\ \Rightarrow\alpha,\beta \in \mathbb{Z}\\ \text{ where } \mathbb{Z}\subset\mathbb{Q} \end{cases} $$ If $(a,b,c)$ are odd then $\alpha \cdot \beta$ is odd and $\alpha + \beta$ is odd, but you cannot have two integers whose product and sum are odd.

So by contradiction we prove that the equation cannot have rational roots

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Could you elaborate on your IFF statement? –  Christian Blatter Mar 24 '13 at 16:00
    
@ChristianBlatter: See my explanation in the answer –  Abhijit Mar 24 '13 at 16:26
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Aaha I noted this down you cannot have two integers whose product and sum are odd. –  Mr.ØØ7 Mar 24 '13 at 16:35
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The proof is incomplete (or incorrect), since the claim that $\alpha$ and $\beta$ are integers (vs. rationals) is not justified. For further explanation see my second answer (the one using the AC-method). –  Math Gems Mar 25 '13 at 0:47
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@MathGems: I thought it was obvious. I have added few more lines in support of this. –  Abhijit Mar 25 '13 at 3:13
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You should think about factoring the polynomial rather than finding its roots. If $ax^2 + bx + c$ has a rational root, then its other root must also be rational, and then it factors in some way like this: $$ax^2 + bx + c = (Ax + B)(Cx + D) \quad A,B,C,D \in \mathbb{Z}$$ Then $a = AC$ and $c = BD$, so if both are odd, then all of $A,B,C,D$ are odd. But then we also have $b = AD + BC$, which is the sum of odd integers, and therefore is even.

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Hint $\ $ By the Rational Root Test, any rational root is integral, hence it follows by

Theorem Parity Root Test $\ $ A polynomial $\rm\:f(x)\:$ with integer coefficients has no integer roots if its constant coefficient and coefficient sum are both odd.

Proof $\ $ The test verifies that $\rm\ f(0) \equiv 1\equiv f(1)\ \ (mod\ 2)\:,\ $ i.e. that $\rm\:f(x)\:$ has no roots modulo $2$, hence it has no integer roots. $\ $ QED

This test extends to many other rings which have a "sense of parity", i.e. an image $\cong \Bbb Z/2,\:$ for example, various algebraic number rings such as the Gaussian integers.

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The lead coefficient is not necessarily $1$, so how does the Rational Root Test tell us that rational roots are integral? –  robjohn Mar 26 '13 at 18:32
    
@robjohn There is a big pending edit to this post that will hopefully appear shortly. I've been waiting to see whether or not the accepted answer is edited for corrections. Many of the answers have gaps, so I plan to discuss the matter in general. –  Math Gems Mar 26 '13 at 18:44
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Suppose that $a,b,c$ are odd. $ax^2+bx+c=0$ has rational roots iff the discriminant is the square of an integer. That is, there is an integer $d$ so that $d^2=b^2-4ac$. Since $a,b,c$ are odd, $d$ must also be odd.

Note that the right hand side of $$ (b-d)(b+d)=4ac\tag{1} $$ has exactly two factors of $2$. However, since $b$ and $d$ are both odd, $2d\equiv2\pmod{4}$ and so one of $b-d$ and $b+d$ is $0\bmod{4}$ and the other is $2\bmod{4}$. Thus, the left hand side of $(1)$ has at least three factors of $2$. Contradiction.

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You can actually prove this in quite an elementary way without even knowing anything about the roots of the quadratic equation. Suppose, on the contrary, that we have rational root $\dfrac {p}{q}$. Then your equation is equivalent to $ap^2+bpq+cq^2=0$. You have that $a,b,c$ are all odd. I will denote odd as an $O$ and even as an $E$.

This breaks into four cases.

1) if $p=O$ and $q=E$ then you have $O+E+E=O=0$, which is impossible

2) if $p=E$ and $q=O$ then you have $E+E+O=O=0$, which is impossible

3) if $p=O$ and $q=O$ then you have $O+O+O=O=0$, which is impossible

4) if $p=E$ and $q=E$ then you have$E+E+E=E=0$. which could be possible ($0$ is an even number), so we treat this case below

Suppose $p=2k$ and $q=2l$ is a solution, then you have that $(p,q)$ is a solution of the equation if and only if $(k,l)$ is a solution, if $k=2e$ and $l=2f$ then you have that $(p,q)$ is a solution if and only if $(e,f)$ is a solution, proceeding in this way you can see that solution, if it exists, must be of the form $(p,q)=(g2^r,h2^s)$ and this also cannot be a solution beacuse this reduces to one of the first 3 cases.

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Here is a proof that is elementary, i.e. requires no knowledge of modular arithmetic. First we use the AC method to reduce to a monic quadratic, i.e. one with leading coefficient $= 1.$

$$\rm\begin{eqnarray} 0\: =\ f(x)\, = &&\rm\ \, a\,x^2+\,b\,x+c\\ \rm \Rightarrow\ \ 0 = a f(x)\, = &&\rm\! (ax)^2\! + b (ax) + ac\\ \rm \Rightarrow\ \ 0 =\, F(X) = &&\rm\ \ \, X^2 \,+\ \color{#C00}b\,\ X\ \,+ \color{#0A0}{ac},\quad X = ax\end{eqnarray}$$

Suppose $\rm\,f(x_i)=0\,$ for $\rm\:\color{brown}{x_i\in\Bbb Q}.\:$ Then $\rm\,F(X_i)=0\,$ for $\rm\: X_i = a\, x_i\in\Bbb Q.\:$ By the Rational Root Test, the rational roots $\rm\,X_i$ are integers. Since their product $\rm = \color{#0A0}{ac}\:$ is odd, the roots are both odd, so their sum is even. This contradicts: by Vieta, the root sum $\rm\, = -\color{#C00}b\,$ is odd, by hypothesis. Hence $\rm\:\color{brown}{x_i \not\in \Bbb Q}.$

Remark $\ $ This yields a conceptual view of the calculations in Abhijit's answer (which, alas, is incomplete, since it does not justify the reduction from rational to integer roots).

The AC-method generalizes to higher degree polynomials (see the above-linked answer). It is intimately connected to various refinement-based views of unique factorization.

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Recall that the square of an odd number is always equivalent to $1$ modulo $8$.

If $a,b,c$ are all odd, then

$$ b^2 - 4 a c \equiv 5 \pmod 8 $$

and thus, $b^2 - 4ac$ cannot be the square of an integer (and thus cannot be the square of a rational number). Therefore the roots are irrational.

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Now that I've written this, I see a downvoted and a deleted answer that I had overlooked employ the same observation. I assume the reason is presentation, so I'll leave my answer in. Although... I don't understand the issues with the answer that got deleted. –  Hurkyl Mar 25 '13 at 4:10
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$x= \dfrac{-b+ \sqrt{b^2-4ac}}{2a} $ or $\dfrac{-b- \sqrt{b^2-4ac}}{2a}$

If $x=\dfrac{p}{q} \implies b^2-4ac=k^2$, and $k$ is odd. ($odd-even=odd$).

Considering $a,b,c$ odd.

$k^2 \equiv 1 \mod 8$

$b^2 \equiv 1 \mod 8$

$4ac \equiv 4 \mod 8$

$b^2-4ac=-3 \mod 8 \implies 5 \mod 8$, a contradiction.(Either of $a$ or $c$ has to be even), since $k^2 \equiv 1 \mod 8$

$\therefore$ You don't get rational roots when all are odd.

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Would you need a constant $a$ at the front of the left hand side of that equation there? i.e. $a(x-\frac{p_1}{q_1})(x-\frac{p_2}{q_2})$ –  Sp3000 Mar 24 '13 at 14:21
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You asked "Why the downvote?" On which version? We are at version 5, version 1 was not an answer, in later versions which parts were a proof and which parts were making circles around the question was unclear, and now the pre-Aliter part is not useful... All in all, you might want to use a draftbook and to post only when you have a full solution. –  Did Mar 24 '13 at 15:11
    
As I said, please change your modus operandi. –  Did Mar 24 '13 at 16:36
    
Well, my previous answer needed a little modification.Right? Even I showed sum and product can't be both odd. But my explanation was flawed. And which modus operandi are you talking about? –  Inceptio Mar 24 '13 at 17:23
    
I thought that was clear enough but let me repeat it: stop posting half digested "solutions", use a draftbook, post only when you have a full solution. (And as a bonus: use the @ thing when you comment on somebody's comment.) –  Did Mar 25 '13 at 23:23
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If $\frac{p}{q}$ is a root then $a\frac{p^2}{q^2}+b\frac{p}{q}+c=0\Rightarrow ap^2+bpq+cq^2=0$. Now we may assume that $p,q$ are not both even; $a,b,c$ are odd, whence contradiction ($p(ap+bq)=-cq^2$ with $p$ even $c,q$ odd or $q(bp+cq)=-ap^2$ with $q$ even $a,p$ odd or ...).

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