How to Solve a Cubic Equation Using Formula and Steps
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 and Complete Guide
1. What is a cubic equation?
A cubic equation is a polynomial equation of degree 3, meaning the highest power of the variable is 3. It is generally written as ax³ + bx² + cx + d = 0, where a ≠ 0.
- a, b, c, d are constants.
- The equation has at most three roots.
- The graph of a cubic equation is called a cubic function and can have one or two turning points.
2. How do you solve a cubic equation?
You can solve a cubic equation using factoring, the rational root theorem, or the cubic formula depending on its form. Common steps include:
- Check for a common factor and factor it out.
- Use the Rational Root Theorem to test possible rational roots.
- Once one root is found, use polynomial division to reduce it to a quadratic.
- Solve the remaining quadratic using the quadratic formula.
3. What is the formula for solving a cubic equation?
The general cubic formula (Cardano’s formula) solves equations of the form x³ + px + q = 0. The solution is:
- x = ∛(-q/2 + √((q/2)² + (p/3)³)) + ∛(-q/2 - √((q/2)² + (p/3)³))
4. How many roots does a cubic equation have?
A cubic equation always has three roots in total, counting real and complex roots. According to the Fundamental Theorem of Algebra:
- It may have three real roots, or
- One real root and two complex conjugate roots.
5. How do you solve a cubic equation by factoring?
To solve a cubic equation by factoring, first find one root and then factor the polynomial. For example:
- Solve x³ − 6x² + 11x − 6 = 0.
- Test possible rational roots: x = 1 works.
- Factor: (x − 1)(x² − 5x + 6).
- Factor quadratic: (x − 1)(x − 2)(x − 3).
6. What is the Rational Root Theorem for cubic equations?
The Rational Root Theorem states that any possible rational root of ax³ + bx² + cx + d = 0 must be of the form ± (factor of d) / (factor of a). Steps:
- List factors of the constant term d.
- List factors of the leading coefficient a.
- Form all possible fractions and test them.
7. Can a cubic equation have three real roots?
Yes, a cubic equation can have three real roots if its graph crosses the x-axis three times. This happens when the discriminant is positive. For example:
- x³ − 6x² + 11x − 6 = 0
- Solutions are 1, 2, and 3 (all real).
8. What is a depressed cubic equation?
A depressed cubic is a cubic equation with no x² term, written as x³ + px + q = 0. It is obtained by substituting x = y − b/(3a) into ax³ + bx² + cx + d = 0. This simplified form is used when applying Cardano’s formula to solve cubic equations.
9. What is the discriminant of a cubic equation?
The discriminant of a cubic equation determines the nature of its roots. For ax³ + bx² + cx + d = 0, the discriminant is:
- Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²
- If Δ > 0: three distinct real roots.
- If Δ = 0: multiple (repeated) roots.
- If Δ < 0: one real and two complex roots.
10. How do you solve a cubic equation using a calculator?
You can solve a cubic equation using a scientific or graphing calculator by using the polynomial solver or graphing method. Steps:
- Enter coefficients in polynomial mode (degree 3), if available.
- Or graph y = ax³ + bx² + cx + d.
- Use the zero/root function to find x-intercepts.
