The Cubic and Quartic Formulas

Formulas to Fear

In Algebra, we are taught the Quadratic Formula as a failsafe way to solve any quadratic equation. All we had to do was rearrange the quadratic equation so that all the terms were on one side, which made the equation look like$$ax^2 +bx +c = 0$$An equation of this form had potentially two solution:$$x = \frac{-b + \sqrt{b^2-4ac}}{2a}$$and$$x = \frac{-b - \sqrt{b^2-4ac}}{2a}$$Similarly, any third degree (cubic) equation, once arranged to be equal to zero, has three definitive formulaic solutions.A cubic equation arranged to be equal to zero can be expressed as$$ax^3 + bx^2 + cx + d = 0$$The three solutions to this equation are given by the Cubic Formula. The first solution is the one that is certain to be real (all odd degree polynomials have at least one real root) and the other two may or may not be real.

Finally, any fourth order (quartic) equation, once arranged to be equal to zero, which can be expressed as$$ax^4 + bx^3 + cx^2 + dx + e = 0$$has up to four distinct solutions including real and imaginary numbers. The four solutions are given by the Quartic Formula (do not try this at home)

Then the four solutions of the equation are(click on the formula to zoom-in with a new tab)

Don't worry about encountering even longer and more complicated formulas for fifth or sixth degree equations. While they would indeed be the stuff that nightmares are made of, they don't exist. Fifth and higher order equations are sometimes solvable, but there is no catch-all general formula for all solutions (and also not all of those higher order equations are solvable).