Can 20(s - t) Be an Integer in a Quartic Equation with No Real Roots?

[anemone]

Hi MHB,

I don't know how to solve the problem below because I found what we are asked to prove is a bit confusing to me and hence I don't know how to formulate a credible way to solve the problem correctly.

Problem:

Let $s\ne t$ be real numbers. The equation $(x^2+20sx+10t)(x^2+20tx+10s)=0$ has no real roots.

State with reason, if $20(s-t)$ is or isn't an integer.

But one can tell if the quartic equation (which is given in factored form, a product of two quadratic functions) has no real roots, then the discriminant of each quadratic function is less than zero. i.e.

$(20s)^2-4(1)(10t)<0$ and $(20t)^2-4(1)(10s)<0$

Subtracting these two inequalities yields

$20(t-s)(10s+10t+1)<0$ or

$20(s-t)(10s+10t+1)>0$ (*)

By expanding the given equation we have

$x^4+20(s+t)x^3+10(s+2st+t)x^2+200(s^2+t^2)x+100ts=0$ (**)

I spent over an hour trying my best to see what relations that (*) and (**) have but I don't know how to go any further...could someone please help me, please

 

[Opalg]

Re: Is 20(s-t) an integer?

Hi MHB,

I don't know how to solve the problem below because I found what we are asked to prove is a bit confusing to me and hence I don't know how to formulate a credible way to solve the problem correctly.

Problem:

Let $x\ne y$ be real numbers. The equation $(x^2+20sx+10t)(x^2+20tx+10s)=0$ has no real roots.

State with reason, if $20(s-t)$ is or isn't an integer.

But one can tell if the quartic equation (which is given in factored form, a product of two quadratic functions) has no real roots, then the discriminant of each quadratic function is less than zero. i.e.

$(20s)^2-4(1)(10t)<0$ and $(20t)^2-4(1)(10s)<0$

Those two inequalities say that $10s^2<t$ and $10t^2<s$. It follows that both $s$ and $t$ are positive. Take the square root of both sides in the second inequality, to get $\sqrt{10}t<\sqrt s.$ Therefore the point $(s,t)$ (and similarly the point $(t,s)$) must lie above the curve $y = 10x^2$ and below the curve $y = \sqrt{\dfrac x{10}}.$ Now use calculus to find the maximum value of $ \sqrt{\dfrac x{10}} - 10x^2.$ I found that the maximum occurs when $x = (16000)^{-1/3}$ and is equal to $\dfrac3{10(16)^{2/3}} \approx 0.04725.$ This is less than $1/20$, so it appears that $20(s-t)$ cannot be an integer.

 

[anemone]

Re: Is 20(s-t) an integer?

Hi Opalg,

Thank you very much for your guidance in this problem. I appreciate your help!:)

 

[Opalg]

Re: Is 20(s-t) an integer?

My previous answer was only partially correct (in fact, it was wrong). It was correct to say that the point $(s,t)$ has to lie in the interior of the region between the curves $y = 10x^2$ and $y = \sqrt{\dfrac x{10}}$. But it was quite wrong to look at the greatest vertical separation between these curves. What we actually need to do is to maximise the difference $|s-t|$. To do that, we need to find where the point $(s,t)$ is as far as possible from the diagonal $y=x$ (the line where $|s-t| = 0$). The way to do that is to find where the tangent to one of the curves is parallel to the diagonal. So we need to differentiate $y=10x^2$ and put the derivative equal to $1$. That gives $20x=1$. So the point $\bigl(\frac1{20},\frac1{40}\bigr)$ lies on the curve, and its coordinates differ by $1/20$. That is the maximum possible difference for $|s-t|$, and it only occurs on the boundary curves. But the inequalities $10s^2<t$ and $10t^2<s$ are strict, so the point $(s,t)$ must lie in the interior of the region between the curves. For any such interior point we must have $|s-t|<1/20$. It follows that $20(s-t)$ cannot be an integer unless it is $0$. But the problem states that $s\ne t$ (actually, it says $x \ne y$ but since there is no $y$ in the problem I think that must be a misprint for $s\ne t$). Therefore $20(s-t)$ cannot be an integer.

 

[CellCoree]

Hi there,

To answer your question, yes, 20(s-t) is an integer. This is because in order for the discriminant of each quadratic function to be less than zero, the expression (20(s-t)) must be a real number. Since s and t are both real numbers, their difference (s-t) must also be a real number. And since 20 is a real number, the product of 20 and (s-t) must also be a real number, which means that 20(s-t) is an integer.

As for solving the problem, I would recommend approaching it in a different way. Instead of trying to find a relationship between (*) and (**), try to think about what would happen if 20(s-t) was not an integer. For example, if 20(s-t) was a fraction, then the product of (20s)^2 and (20t)^2 would not be less than 4(1)(10t) and 4(1)(10s), respectively. This would result in the discriminant of one of the quadratic functions being greater than or equal to zero, which would contradict the given information that the equation has no real roots.

I hope this helps! Let me know if you have any other questions.