How to Solve Cubic and Quartic Equations
Earlier, you learned to solve linear equations (equations where the highest power is 1) and quadratic equations (equations where the highest power is 2). In this section you’ll learn to solve equations with powers of all possible values. You’ll mainly look at cubic and quartic equations—the method is the same for both.
Solve the equation x3 + 2x2 + x = 0
Solve the equation x4 = 9x2
x4 = 9x2 x4 − 9x2 = 0 x2 (x2 − 9) = 0 x2 (x − 3) (x + 3) = 0 Now you use the zero product property to find the solutions:
Solve the equation x3 − 7x = −6
Now, you’ll have to guess a solution. Begin with x = 1:
Lucky for you, the first solution you guessed was correct. (You often start with 1 when you guess a solution, and this is why).
Now you have to use polynomial long division on the equation with (x − 1). Because the expression lacks the x2-term, you put in an extra space where the x2-term would have been, or put 0 in front of x2 in the long polynomial division. It’s easier to maintain control over the terms this way.
you get the answers x = 2 and x = −3. Put them in the factorization formula a (x − x1) (x − x2) so that the factorization becomes