Finding all Possible Values of $x$ in Triangle ABC

[anemone]

Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle B\ge 90^{\circ}$ and $x$ is a real number satisfying the equation $x^4+ax^3+bx^2+cx+1=0$ where $a=BC,\,b=CA$ and $c=AB$. Find all possible values of $x$.

 

[jvicens]

Hello, thank you for your question. I would like to provide a thorough explanation of the possible values of $x$ in this scenario.

First, let's consider the relationship between the side lengths of a triangle inscribed in a circle. According to the inscribed angle theorem, an inscribed angle is half the measure of the intercepted arc. This means that in triangle $ABC$, $\angle B$ is half the measure of the arc $AC$. Since $AC$ is a semicircle with a radius of 2, its arc measure is $180^{\circ}$, and therefore $\angle B = 90^{\circ}$.

Now, let's consider the given equation $x^4+ax^3+bx^2+cx+1=0$. This is a quartic equation, meaning it has four possible solutions for $x$. However, we can use the rational root theorem to narrow down the possible values of $x$.

The rational root theorem states that if a polynomial with integer coefficients has a rational root, it will be in the form of $\frac{p}{q}$ where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. In this case, the constant term is 1 and the leading coefficient is 1, so the possible rational roots are $\pm1$. This means that the possible values of $x$ are $\pm1$.

Now, let's consider the triangle $ABC$ again. We know that $\angle B$ is a right angle, so by the Pythagorean theorem, $AB^2+BC^2=AC^2$. Substituting in the values of $a$, $b$, and $c$, we get $x^8+2x^6+3x^4+2x^2+1=0$. This equation has the same possible roots as the given equation, $\pm1$. However, we need to check if these values actually satisfy the equation.

Plugging in $x=1$, we get $1+2+3+2+1=0$, which is not true. Similarly, plugging in $x=-1$, we get $1-2+3-2+1=1$, which is also not true. This means that there are no real solutions for $x$ in this scenario.

In conclusion, the only possible values of $