Solving Quartic Equations Revision Content

A quartic equation is a polynomial equation of degree 4. The general form is ax^4 + bx^3 + cx^2 + dx + e = 0, where ‘a’ is non-zero.
Quartic equations have up to 4 real roots and can have 2 or 0 real roots with the others being complex.

The simplest way to solve a quartic equation is by factoring, if possible. Factoring the equation can simplify it to a product of quadratic equations or lower, which can be solved easily.
Check if the equation can be factored into two quadratic equations, or a squared binomial and a quadratic equation.

Synthetic division can be used to simplify the quartic equation if one of its roots can be guessed or known beforehand.
After performing synthetic division by the known root, a cubic equation is obtained which is relatively easier to solve.

If the quartic equation can be factored into two quadratic equations, solving each quadratic equation with the quadratic formula will give the roots of the quartic equation.
Quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a

The Ferrari method is a mathematical approach to find the roots of quartic equations.
It involves completing the square and manipulating the equation until it can be factored into two quadratic equations.
The Ferrari method is more complex and should be used for quartic equations that cannot be factored or solved using synthetic division.

If the discriminant of any resulting quadratic equation is negative, it has two complex roots. This can occur even if the quartic equation itself only contains real coefficients.
Remember that if a + bi is a root, so is its conjugate a - bi.

The roots can be real or complex. If the quartic equation has real coefficients, complex roots always come in conjugate pairs.
A quartic equation has at least one real root. This is a consequence of the fundamental theorem of algebra.

Sketching the graph of the quartic function can also give a visual representation of the roots, as they correspond to the x-intercepts of the graph.
A clever choice of substitution can often simplify the solution process, especially when recognizing patterns within the equation.
Understanding the relationships between roots and coefficients given by the coefficients of the Vieta’s formulae can also be very useful when solving quartic equations.