How to Solve Cubic Equations Easily: Strategies and Examples
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: Methods, Steps & Online Calculator
1. What is a cubic equation?
A cubic equation is a polynomial equation of degree three. This means the highest exponent of the variable in the equation is 3. Its standard form is written as ax³ + bx² + cx + d = 0, where 'a', 'b', 'c', and 'd' are coefficients (constants), 'x' is the variable, and 'a' cannot be zero.
2. What are the 'roots' or 'solutions' of a cubic equation?
The roots of a cubic equation are the values of the variable (x) that make the equation true. In other words, when a root is substituted for 'x' in the equation, the expression evaluates to zero. Every cubic equation has exactly three roots, although they may not all be distinct or real numbers.
3. How do you solve a cubic equation step-by-step without using a calculator?
To solve a cubic equation manually, you can use the factorisation method, which is common in the CBSE syllabus:
- Step 1: Ensure the equation is in the standard form ax³ + bx² + cx + d = 0.
- Step 2: Find one integer root using the Rational Root Theorem or by trial and error. Test factors of the constant term 'd'. Let's say you find a root, x = k.
- Step 3: Since x = k is a root, (x - k) must be a factor of the polynomial. Use polynomial long division to divide the cubic polynomial by (x - k).
- Step 4: The result will be a quadratic equation. Solve this quadratic equation using the quadratic formula or by factoring to find the remaining two roots.
4. How many roots does a cubic equation have, and can they be complex?
According to the Fundamental Theorem of Algebra, a cubic equation always has three roots. The nature of these roots can vary:
- All three roots can be distinct real numbers.
- There can be one real root and two other real roots that are identical (a repeated root).
- There can be one real root and a pair of complex conjugate roots (e.g., p + qi and p - qi).
A cubic equation with real coefficients cannot have only one or three complex roots; they must appear in conjugate pairs.
5. How does an online cubic equation solver find the roots?
An online cubic equation solver automates the process of finding roots. When you input the coefficients (a, b, c, d), the calculator applies either:
- Numerical methods to approximate the roots with high precision.
- A direct algebraic formula like Cardano's method, which is a complex but exact formula for finding all three roots.
The solver instantly provides all real and complex roots, saving you from lengthy manual calculations.
6. What is the difference between a cubic equation and a cubic polynomial?
The key difference lies in the presence of an equality. A cubic polynomial is an expression of the form ax³ + bx² + cx + d. A cubic equation is a statement that sets this polynomial equal to zero: ax³ + bx² + cx + d = 0. The goal with a polynomial is often to evaluate it, whereas the goal with an equation is to solve for the values of 'x' that satisfy the equality.
7. Can a cubic equation with real coefficients have only imaginary roots? Why or why not?
No, a cubic equation with real coefficients cannot have three imaginary or complex roots. This is because complex roots of polynomials with real coefficients always occur in conjugate pairs (e.g., if a + bi is a root, then a - bi must also be a root). Since a cubic equation has three roots, you can have one pair of complex conjugates, but the third root must be a real number to complete the set.
8. Where are cubic equations used in real-world scenarios?
Cubic equations are important for modelling complex behaviours in various fields:
- Engineering: For designing the shapes of structures, like arches or roller coaster tracks, and in fluid mechanics.
- Physics: To describe the state of certain thermodynamic systems and in orbital mechanics.
- Computer Graphics: For creating smooth, three-dimensional curves and surfaces, known as Bézier splines.
- Economics: To model complex cost, revenue, and profit functions where returns are not linear.
9. Why is factoring a common method for solving cubic equations in school instead of using the complex general formula?
Factoring is preferred in the high school curriculum because it builds and reinforces fundamental algebraic concepts. While a general formula (Cardano's method) exists, it is extremely complicated and often requires working with complex numbers even to find real solutions. The factoring method effectively tests a student's understanding of the Factor Theorem, polynomial division, and solving quadratic equations, which are crucial skills for higher mathematics.
10. What happens to a cubic equation if its leading coefficient becomes zero?
If the leading coefficient 'a' in the equation ax³ + bx² + cx + d = 0 becomes zero, the term ax³ disappears. The equation is no longer a cubic equation. It reduces to a quadratic equation: bx² + cx + d = 0. This is an equation of degree two, which is solved using different methods, such as the quadratic formula.
