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In algebra, a cubic equation in one variable is an equation of the form in which a is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation.
- Overview
- Solving Cubic Equations without a Constant
- Finding Integer Solutions with Factor Lists
- Using a Discriminant Approach
- Practice Problems and Answers
In a cubic equation, the highest exponent is 3, the equation has 3 solutions/roots, and the equation itself takes the form .
While cubics look intimidating and unlike quadratic equation is quite difficult to solve, using the right approach (and a good amount of foundational knowledge) can tame even the trickiest cubics. You can try, among other options, using the
Check whether your cubic contains a constant (a value).
Cubic equations take the form .
However, the only essential requirement is , which means the other elements need not be present to have a cubic equation.
If your equation does contain a constant (a
value), you'll need to use another solving method.
, you do not have a cubic equation.
Ensure your cubic has a constant (a nonzero value).
If your equation in the form has a nonzero value for , factoring with the quadratic equation won't work. But don’t worry—you have other options, like the one described here!
In this case, getting a
on the right side of the equals sign requires you to add
, you can't use the quadratic equation method.
Find the factors of and .
Write out the values of , , , and .
For this method you’ll be dealing heavily with the coefficients of the terms in your equation. Record your , , , and terms before you begin so you don't forget what each one is.
Don't forget that when an
variable doesn't have a coefficient, it's implicitly assumed that its coefficient is
Calculate the discriminant of zero using the proper formula
The discriminant approach to finding a cubic equation's solution requires some complicated math, but if you follow the process carefully, you'll find that it's an invaluable tool for figuring out those cubic equations that are hard to crack any other way. To start, find (the discriminant of zero), the first of several important quantities we'll need, by plugging the appropriate values into the formula .
Practice Problems and Answers to Solve a Cubic Equation
How do you solve a simple cubic equation?
If you only have x³ in an equation, you can isolate it and find the cube root of both sides. This only works with really simple equations, though—factoring is the best way to solve more complex equations.
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The cubic equation formula expresses the cubic equation in Mathematics. An equation with degree three is called a cubic equation. The nature of roots of all cubic equations is either one real root and two imaginary roots or three real roots.
Aug 17, 2023 · Cubic Equation Calculator solves cubic equations or 3rd degree polynomials. Uses the cubic formula to solve third order polynomials for real and complex solutions.
- Determine the roots of the cubic equation 2x3 + 3x2 – 11x – 6 = 0. Solution. Since d = 6, then the possible factors are 1, 2, 3 and 6. Now apply the Factor Theorem to check the possible values by trial and error.
- Find the roots of the cubic equation x3 − 6x2 + 11x – 6 = 0. Solution. x3 − 6x2 + 11x – 6. (x – 1) is one of the factors. By dividing x3 − 6x2 + 11x – 6 by (x – 1),
- Solve x3 – 2x2 – x + 2. Solution. Factorize the equation. x3 – 2x2 – x + 2 = x2(x – 2) – (x – 2) = (x2 – 1) (x – 2) = (x + 1) (x – 1) (x – 2) x = 1, -1 and 2.
- Solve the cubic equation x3 – 23x2 + 142x – 120. Solution. First factorize the polynomial. x3 – 23x2 + 142x – 120 = (x – 1) (x2 – 22x + 120) But x2 – 22x + 120 = x2 – 12x – 10x + 120.
1 day ago · The cubic formula is the closed-form solution for a cubic equation, i.e., the roots of a cubic polynomial. A general cubic equation is of the form z^3+a_2z^2+a_1z+a_0=0 (1) (the coefficient a_3 of z^3 may be taken as 1 without loss of generality by dividing the entire equation through by a_3).
A cubic equation is an equation which can be represented in the form \(ax^3+bx^2+cx+d=0\), where \(a,b,c,d\) are complex numbers and \(a\) is non-zero. By the fundamental theorem of algebra , cubic equation always has \(3\) roots, some of which might be equal.
