Roots of quartic equations

Quartic equations, also known as fourth-degree equations, follow the general form ax^4 + bx^3 + cx^2 + dx + e = 0, where a, b, c, d and e are constants.
The roots of a quartic equation denote the x-value(s) at which the function equals zero. This means that the drawn curve of a quartic equation would intersect the x-axis at its roots.
Quartic equations can have 0, 1, 2, 3, or 4 real roots. The number of roots depends on the structure of the equation and can be found using techniques like factoring, using the quadratic formula, or applying the rational root theorem.
The Rational Root Theorem states that any rational root of an equation ax^n +bx^(n-1)…+k = 0, can be written in the form of ± p/q, where p is a factor of the constant ‘k’ (last term in the equation) and q is a factor of the leading coefficient ‘a’.
Quartic equations can be factored into two quadratic equations if they can be written in the form of ax^4 + bx^2 + c = 0. This method provides a simpler approach than trying to solve the entire quartic equation directly.
Quartic equations may have complex roots, and these would correspond to points on the graph of the quartic function that intersect the x-axis in the complex plane.
The fundamental theorem of algebra confirms there will always be four roots for a quartic equation, though these roots may be real, complex, and/or repeated.
Quartic equations can also be solved using Ferrari’s method, which involves turning the quartic into a quadratic by perfecting the cube.
Sometimes quartic equations can be simplified and made easier to handle through substitution. This may involve assigning the value of x^2 (denoted as some variable) to transform the quartic equation into a quadratic equation.
Additionally, Descartes’ Rule of Signs can be invaluable for predicting the possible number of positive and negative real roots of a quartic equation, based on the changes in the sign of the coefficients.