What is Cubic Equation?
An equation in algebra is a mathematical assertion that comprises an equal to (=) symbol amidst two algebraic expressions with the same value. At least one or more variables are utilized in the most fundamental and typical algebraic equations. For example, 5e+8 = 4 is an equation wherein the expression 5e+8 is equal to 4. You can consider an equation to be a set of weights or dumbbells with varying amounts on the left and right sides. Your job, thus, is to balance the two weights or sides to answer the problem.
It is noteworthy to highlight that an intrinsic and indispensable part of an algebraic equation is the unknown variable that can be either deemed, x, y, or z. If you want to solve an equation, you have to utilize identical procedures on both sides to determine the value of the unknown integer. Also, the BODMAS, Brackets, Order, Division, Multiplication, Addition, and Subtraction- method has to be followed to solve an equation.
Now that we have learned about the fundamentals of an equation let us look into the concept of a cubic equation. There are three types of equations, and the cubic equation is one of them. Before answering the question – of how to solve a cubic equation – let us understand what is a cubic equation.
A cubic equation is an equation in which the maximum power of the variables in the equation is 3. The typical form of a cubic equation is ax3+bx2+cx1+dx0=0, where a, b, c, d are constants and ‘a’ is not equal to ‘0’. Moreover, a, b, c, d are coefficients of x3,x2,x1,x0 respectively. Now that we have briefly touched upon an equation with cubic formula algebra, let us talk about the cubic equation solver.
\[x =\sqrt[3]{(\frac{-b^{3}}{27a^{3}} +\frac{bc}{6a^{2}} - \frac{d}{2a}) + \sqrt{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d^{2}}{2a})^{2} + (\frac{c}{3a} - \frac{b^{2}}{9a^{2}})^{3}}} + \sqrt[3]{(\frac{-b^{3}}{27a^{3}} +\frac{bc}{6a^{2}} - \frac{d}{2a}) - \sqrt{(\frac{-b^{3}}{27a^{3}} + \frac{bc}{6a^{2}} - \frac{d^{2}}{2a})^{2} + (\frac{c}{3a} - \frac{b^{2}}{9a^{2}})^{3}}} - \frac{b}{3a}\]
How to Solve Cubic Equations?
Let us finally delve into how to solve the cubic equation. In the conventional sense, we can solve a cube equation by reducing it to a quadratic equation and opting for the factoring method or the quadratic formula. There may be three real roots in a cubic equation, just like a quadratic equation which can possess at least two real roots. Likewise, a cubic equation necessarily owns at least one real root, unlike a quadratic equation with no real solutions at a time. There is at least one real root in a cubic equation, and the other two may be imaginary.
The first step of a cubic equation solver entails recognizing it in a standard form. For instance, if the haphazard equation is 5x3+2x2–5x=3,then we have to arrange it in a standard format and write it as 5x3+2x2–5x–3=0. Then, we can solve it in the following ways-
Roots of Cubic Equation Method
To begin with, let us assume that we have found the roots of f(x) = x3–4x2–6x+5=0. Then, by opting for the roots of the cubic equation method, we have to find the possible factors in the equation. The first step is to find the factors of the constant term, and then we have to put those values and examine whether or not they are satisfying enough. Since d=5, then the possible factors are 1 and 5
Step 1: First, use the factor theorem to check the possible values by the trial-and-error method.
f(1)=1–4–6+5≠0
f(–1)=1–4+6+5≠0
f(5)=125–100–30+5=0
We find that the root is 5.
Step 2: Find the other roots either by inspection or by the long division method.
x3–4x2–6x+5=0
(x–5)(x2+x–1)=0
So, the roots are x=5,\[\frac{-1+\sqrt{5}}{2}\],\[\frac{-1-\sqrt{5}}{2}\]
Example: Check whether (2x–3) is a factor of (x+2x3–9x2+12).
Let p(x)=2x3–9x2+x+12
p(x)=2x3–9x2+x+12 and g(x)=2x–3
Now, g(x)=0
⇒2x–3=0
⇒2x=3⇒x=3/2
By factor theorem, g(x)will be a factor of p(x), if p(3/2)=0
Now, p(3/2)={2×(3/2)3–9×(3/2)2+3/2+12}
={(2×\[\frac{27}{8}\])–(9×\[\frac{9}{4}\])+\[\frac{3}{2}\]+12}
=0
Since p(3/2)=0, so, g(x) is a factor of p(x)
Thus, we can examine the roots of cubic equation method of solving an equation.
Graphical Method of Solving a Cubic Equation
This method of cubic equation solver involves breaking down a precise drawing of the cubic equation. A solution of the equation is the point where the point or points crosses the X-Y axis. The number of real solutions for cubic equations is the number of times the graph crosses the X-Y axis.
In conclusion, we can meticulously analyze the question – how to find the roots of a cubic equation – and the various other nitty-gritty of the investigation. If you find it hard to solve a cubic equation manually, you can also opt for a cubic equation calculator to help you deal with the problem. It is integral to grasp the many nitty-gritty of cubic roots because they are used in many mathematical topics. Algebraic equations form the premise of mathematics, and everyone should be equipped with the basic knowledge to solve and understand them.
FAQs on Cubic Equation Solver
1. What is the history associated with the mathematical nuances of a cubic equation?
The history of cubic equations dates back to the 20th to 16th centuries BCE. Studies show that Babylonians, Greeks, Indians, and Egyptians were aware and even adept in solving cubic formula algebra. The oldest and the most straightforward cubic equation problem of doubling the cube was first solved in the 4th century BCE by Hippocrates. He was able to reduce the problem to finding two mean proportionals between one line and another line of twice the length.
2. What is a polynomial equation?
An equation that comprises variables, exponents, and coefficients is termed a polynomial equation. We can manually solve a polynomial equation by factoring it in terms of degree and variables. You can opt for a polynomial equation calculator to further analyze the solving of a polynomial equation.
