Verify that $abc^2x^2+c(3a^2+b^2)x+3a^2-ab+b^2=0$ has real roots

Question

As the title states, the goal is to verify that the quadratic equation: $abc^2x^2+c(3a^2+b^2)x+3a^2-ab+b^2=0$ has real roots. This problem comes from an interschool mathematics contest for High-schoolers, here's my (brute force) attempt:

Since we need to verify that the roots are rational, we need to prove that the discriminant of this equation (which I'll denote with $D$) is greater than $0$. In other words:

$D=B^2-4AC>0$ where $B=c(3a^2+b^2)$, $A=abc^2$ and $C=3a^2-ab+b^2$ Therefore:

$D=[c(3a^2+b^2)]^2-4(abc^2)(3a^2-ab+b^2)$

$D=[c^2(9a^4+6a^2b^2+b^4)]+[c^2(-12a^3b+4a^2b^2-4ab^3)]$

$D=c^2(9a^4+10a^2b^2+b^4-12a^3b-4ab^3)$

I'm not quite sure where to proceed from here. And this certainly does not prove that the roots are real as this expression can be negative. Is there a way to proceed from here? Are there any better or alternative ways to answer this? Please share your approaches!

Answer 1

$$ p(x) = ab{c}^{2}{x}^{2}+c \left( 3\,{a}^{2}+{b}^{2} \right) x+3\,{a}^{2}-ab+{b}^{2} . $$ Let's try to factor this with the idea of finding a factor $Ax+B$ where $A$ is a divisor of $abc^2$ and $B$ is a divisor of $3a^2-ab+b^2$. Since $3a^2-ab+b^2$ cannot be factored over the rationals, we try without factoring it. (If this doesn't work, we have to do something harder.) The possible factors of $p(x)$ are $Ax+B$ where $$ A \in \big\{1, a, b, ab, c, ac, bc, abc, c^2, ac^2, bc^2, abc^2\big\} , \\ B \in \big\{1, -1, (3a^2-ab+b^2), -(3a^2-ab+b^2)\big\} . $$ Trying these four candidates for $B$ until one works, we arrive at: $$ p(x)= \left( cx+1 \right) \left( abcx+(3\,{a}^{2}-ab+{b}^{2})\right) $$

Answer 2

It seems your approach is the most straightforward, and you are almost done. Observe that WLOG we may assume $a, b \ge0$, and we, by AM-GM, have:

$$9a^4+10a^2b^2+b^4 \ge (6a^4+6a^2b^2)+(4a^2b^2+b^4) \ge 12a^3b +4ab^3.$$

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EDIT: Another easy way is: $D=c^2(3a^4+6(a^2-ab)^2+(b^2-2ab)^2),$ not using the AM-GM inequality.

Answer 3

Since this problem was written for high-school contestants, I also have my doubts as to how many of them would spot the factorability of the discriminant. I decided to take the approach of writing the quadratic polynomial in "vertex form", thus $$ abc^2·x^2 \ + \ c·( \ 3a^2 + b^2 \ )·x \ + \ ( \ 3a^2 - ab + b^2 \ ) $$ $$ = \ \ abc^2 · \left[ \ x^2 \ + \ \frac{\overbrace{3a^2 \ + \ b^2}^{K}}{abc}·x \ \right] \ + \ ( \ K \ - \ ab \ ) $$ $$ = \ \ abc^2 · \left( \ x \ + \ \frac{K}{2abc} \ \right)^2 \ + \ \left( \ K \ - \ ab \ - \ \frac{K^2}{4a^2b^2c^2}·abc^2 \ \right) $$ $$ = \ \ abc^2 · \left( \ x \ + \ \frac{K}{2abc} \ \right)^2 \ + \ \left( \ K \ - \ ab \ - \ \frac{K^2}{4ab} \ \right) \ \ , $$ with $ \ K \ > \ 0 \ $ and (necessarily) $ \ ab \ \neq \ 0 \ \ . \ $ (I am presuming, as it is not stated, that $ \ a \ , \ b \ , \ c \ $ are all real.)

The only sign dependence of significance in this expression is that of the recurrent factor $ \ ab \ \ : \ $ if $ \ ab \ > \ 0 \ \ $ (the parabola "opens upward"), then this quadratic polynomial has real zeroes if $ \ \left( \ K \ - \ ab \ - \ \frac{K^2}{4ab} \ \right) \ \le \ 0 \ \ ; \ $ if $ \ ab \ < \ 0 \ \ $ (the parabola "opens downward"), then the direction of this "coefficients" inequality is reversed. With $ \ ab \ > \ 0 \ , \ $ the term $ \ \frac{K^2}{4ab} \ $ is also positive, so it remains to determine whether $ \ K \ \le \ ab \ + \ \frac{K^2}{4ab} \ \ . \ $ Since both the numerator and denominator in these ratios are positive, we may apply the AM-GM inequality to show that indeed $$ \frac{ ab \ + \ \frac{K^2}{4ab}}{2} \ \ \ge \ \ \sqrt{ab · \frac{K^2}{4ab} } \ \ \Rightarrow \ \ ab \ + \ \frac{K^2}{4ab} \ \ \ge \ \ 2·\sqrt{ \frac{K^2}{4} } \ = \ K \ \ . $$ Hence, $ \ \left( \ K \ - \ ab \ - \ \frac{K^2}{4ab} \ \right) \ \le \ 0 \ $ for this case. [This portion ends up being similar to Reza Rajaei's argument.] With $ \ ab \ < \ 0 \ \ , \ \ K \ - \ ab \ - \ \frac{K^2}{4ab} \ $ is equivalent to a sum of three positive terms, hence $ \ \left( \ K \ - \ ab \ - \ \frac{K^2}{4ab} \ \right) \ \ge \ 0 \ \ . $

Therefore, the polynomial in question always has at least one real zero. (Other than that we must have $ \ c \ \neq \ 0 \ \ , \ $ the value of $ \ c \ $ is of no consequence for this conclusion.)