How to solve a cubic equation easily

Tracing back to the 1500s, the method for solving cubic equations was founded by two Italian mathematicians. Algebra is in itself a challenging subject, but it is in solving cubic equations that most of you falter. Cubic Equations or what is colloquially known, as a third-degree equation can be quite challenging to solve owing to the fact that you need to solve it in steps. If you make a mistake in determining the value in one of the steps, the answer of the entire sum may come out to wrong thus putting all your efforts to waste.

Here are some of the difficulties that most of the students face while solving a cubic equation.

• Poor mathematical foundations - If you do not have a fundamental knowledge of algebra, solving a cubic equation can prove to be nearly impossible. You first need to know the key concepts well to solve complex equations.
• Using the wrong approach – Cubic equations can be solved using three different methods. You need to determine the approach on the basis of the values that are given to you. Even if you know the concepts and formulas correctly, many of you do not know the right approach.
• Making silly mistakes – The arch nemesis of any mathematical problem is making silly mistakes like using the wrong values or taking the wrong signs while considering the values in the equation.

Sure, cubic equations can be really a nightmare. But with the right approach and a good foundation of the key concepts, even the most confounding of the cubic equations can be solved methodically.

To make sure that you are entirely clear about the various methods of cubic, here is everything that you need to know about solving roots in a cubic expression. You can even get algebra homework help for any algebric equations.

Method 1: Quadratic Formula

The first method is for an equation that is similar in form to the equation structure ax³ + bx² + cx + d = 0. Generally, this approach is used when the equation does not have a constant or the constant is considered to be zero. This approach is relatively more straightforward where all you need to do is put the values in the equation to find the value of x. {fSince every term in the equation has a {\dvariable in it, you must find the factor to simplify it. Once simplified, put it in the “quadratic formula {\displaystyle ax^{2}+bx+c}equals to zero” framework to find out the value of x. The value of x is always considered in both the negative and positive signs. 

Method 2: Finding integers with factor lists

When the cubic has a constant, the approach adopted to solve is a tad different than how it is solved when the constant is not provided. To start with, first, find the factors of {\displaystyle a} the coefficient of the {\displaystyle x^{3}}term x³ and {\displaystyle d}the constant at the end of the equation or “d”. Now after you are done factorising the values, {make a list of the values (which will include {fractions as well as whole numbers). Since the answer should be an integer, consider the whole numbers as your cubic equation will be either one of the whole numbers in this list. To see which whole number is the right one, you can plug the values manually in the quadratic equation which ones equal zero. You can also use synthetic division to find out the correct answer. 

Method 3: The “Discriminant” Approach

Δ0 = b² - 3ac

{This method of finding a cubic equation's solutions deals with the coefficients of the terms in of an equation. Before you start, make sure you record the correct values of a, b, c and d. Keep in mind {\displaythat in an equation if x does not have a value, then the coefficient for the same is considered to be 1. {\displaystyle \triangle _{0}=b^This approach is a little tedious as it involves some complicated calculations, but if you get a good grasp over this, it will take you a few minutes to solve the otherwise time-consuming equations. You need to start with solving and trying Δ0 and then use the values into the formula b³ – 3ac for Δ1.

If the discriminant is a positive integer, then the equation will have three real solutions, and if the discriminant is a negative one, then the equation will have only one answer. In case it is zero, then the equation will have one or two real solutions.

Note: Any cubic equation must have at least one real solution because the graph will always cross the x-axis once.)

Now the question is, How to get better at solving cubic equations?

Knowing the methods theoretically is very different from applying them. Now that you are clear about the various ways of solving a cubic equation, here are some tips that can help you be more efficient at solving a cubic equation.

• Have a clear understanding of the problem

Many a time, students start working on a sum without reading the entire question properly and miss out essential values. So be very careful while reading the question. Make a note of all the value given to you and determine the approach to it accordingly.  

• Break it up

Since cubic equations can be lengthy and perplexing, divide the sum into parts and work on them one at a time. Do not try to find the variables all together. Instead, solve them out one at a time. Working on an equation step-by-step will also help you spot errors easily so that you do not have to solve the entire sum all over again.

• Do not get up in between

Quite often students make the mistake of getting up while solving a sum. When it comes to solving algebraic equations, especially cubic equations, this can prove to be a terrible mistake. A digression could make you forget where you were and as a result hinder the process.

• Look for the constant

As you already know, there are three different methods that can be used to solve a cubic equation. However, every approach is different from another depending on the values available. So when you are asked to solve a cubic, look for the constant first to determine which method to follow. Take note of the relationship between the variables as well.

• Take note of anything you feel is important

While reading the word problem, get into the practice of underlining the things that you find important like the keywords or the variables. This will help you to refer them quickly and will make it easier to go back and check the values are right or not.

• Embrace digital learning

In today’s digital world, why not make optimal use of the various technological tools that are available in the market? If you cannot understand how to solve a sum from the textbook and your teacher cannot explain to it either, resort to the power of the internet. There are several online tutorials that can help you understand algebra better and therefore master the art that solving cubic equations is.

Above anything else, algebra especially a cubic equation is all about practice on a consistent basis. So practice cubic problems every day to your solving methods and get used to the sums. If you cannot understand a sum, do not think twice before asking for help.

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