This questions is related to the signs of the coefficients of quadratic polynomials.however doesn't ask about the coefficients themselves.

Question

Suppose a,b,c are three real numbers, such that the quadratic equation x²-(a+b+c)x+(ab+bc+ac)=0,has roots of the form α±iβ, where α>0 β≠0 are real numbers (i=sqrt(-1)). Show that (I) the numbers a,b,c are all positive. (II) the numbers √a,√b,√c form the sides of a triangle.

I tried to look at the roots of the quadratic equation by using the quadratic formula.

[(a+b+c)±√((a+b+c)²-4(ab+bc+ac))]/2 = (a+b+c)/2 ± √D/2

Here by comparison we have α=(a+b+c)/2 and β=√D/2 Now the discrimant has to be negative . Hence we have (a+b+c)²<4(ab+bc+ac)

(a²+b²+c²)<2(ab+bc+ca) Now a+b+c must me positive since It corresponds to α in α±iβ which is given to be positive..Now ab+bc+ac is also positive since it is greater than sums of the perfect squares of 3 numbers.so a+b+c ,ab+bc+ac postive and twice of ab+bc+ac is greater than a²+b²+c². This is all where I got.. I also understood what they mean by √a,√b,√c form triangle. It means that sum of any pair of square roots must exceed the the third square root. But I don't understand how it is

Answer 1

To prove $a, b, c > 0$, you can assume one or two of them are not positive (it is clear that one at least is positive since $a + b + c > 0$) and try to contradict : $a^2 + b^2 + c^2 < 2 ab + 2 bc + 2 ac$.

For instance, if $a, b > 0$ and $c \leq 0$, you have : $a^2 + b^2 + c^2 < 2 ab - 2 |bc| - 2 |ac|$, which is equivalent to $(a-b)^2 + c^2 + 2 |bc| + 2 |ac| < 0$, impossible. The other possibility can be treated the same way.

For the second part, you can assume $0 < a \leq b \leq c$ for instance and all you have to prove is $\sqrt{a} + \sqrt{b} \geq \sqrt{c}$. Since everything is positive, this is equivalent to $a + b + 2 \sqrt{ab} \geq c$, and then to $4 ab \geq c^2 + a^2 + b^2 - 2 ac - 2 bc + 2 ab$... which is exactly the inequality you already have. (Note that, since it is even a strict inequality, the triangle is a "real" triangle, i.e. it is not a flat one.)