- #1
Siron
- 150
- 0
Hello,
Given the quartic:
$$16x^4-40ax^3+(15a^2+24b)x^2-18abx+3b^2 = 0$$
where $a,b$ are certain real constants. My question is if there is a (simple) condition on $a$ and/or $b$ such that the quartic has at least one real root.
Since the quartic has real coefficients the only possibilities for the roots are: 2 reals and 2 complex roots, 4 complex roots and 4 real roots. I found an article by E.L.Rees (Graphical discussion of the roots of a quartic equation) that describes the criteria regarding the nature of the roots for a general reduced quartic $x^4+qx^2+rx+s = 0$. It it not difficult to write the given quartic in this reduced form so this gives me quite a good idea whenever there is at least one real root (2 reals and 2 complex or 4 reals) in function of the discriminant and some conditions on the coefficients.
The expression for the discriminant of the given quartic is:
$$\frac{27}{4}b^6+\frac{27297}{1024}a^4b^4-\frac{729}{32}a^2b^5+\frac{10125}{8192}a^8b^2-\frac{6075}{512}a^6b^3$$
which in my opinion is not that nice to work with.
To summarize, using the discriminant I can derive the necessary conditions on the coefficients $a$ and $b$ to have at least one real root. However the expressions can be quite hard/ugly to work with so I'm wondering if there is no better or more efficient method here.
Thanks in advance!
Given the quartic:
$$16x^4-40ax^3+(15a^2+24b)x^2-18abx+3b^2 = 0$$
where $a,b$ are certain real constants. My question is if there is a (simple) condition on $a$ and/or $b$ such that the quartic has at least one real root.
Since the quartic has real coefficients the only possibilities for the roots are: 2 reals and 2 complex roots, 4 complex roots and 4 real roots. I found an article by E.L.Rees (Graphical discussion of the roots of a quartic equation) that describes the criteria regarding the nature of the roots for a general reduced quartic $x^4+qx^2+rx+s = 0$. It it not difficult to write the given quartic in this reduced form so this gives me quite a good idea whenever there is at least one real root (2 reals and 2 complex or 4 reals) in function of the discriminant and some conditions on the coefficients.
The expression for the discriminant of the given quartic is:
$$\frac{27}{4}b^6+\frac{27297}{1024}a^4b^4-\frac{729}{32}a^2b^5+\frac{10125}{8192}a^8b^2-\frac{6075}{512}a^6b^3$$
which in my opinion is not that nice to work with.
To summarize, using the discriminant I can derive the necessary conditions on the coefficients $a$ and $b$ to have at least one real root. However the expressions can be quite hard/ugly to work with so I'm wondering if there is no better or more efficient method here.
Thanks in advance!