How to Solve Higher-Order Equations Easily
LikeMaths isn’t the easiest subject to pass, and the fact that it’s one of the subjects where most students struggle proves it. For many students, maths lessons become a nightmare that feels impossible to succeed without extra help from a private maths tutor.
Among the topics that we learn in secondary education, equations are one of the hardest things to master. Many students find them incredibly difficult, barely passing exams, and then forget them forever. My personal experience with higher-order equations was exactly the same at first, but with a good study plan and consistent practice, equations will feel just like puzzles.
Equations will always be in your life if you want a career in STEM. I know, this can be stressful at first but, to avoid the stress, and, more importantly, to ensure you pass, it is important to build a solid foundation. There are different types of equations, and the most common ones are the quadratic equations, if you don't know how to solve quadratic equations yet, I recommend you to start with that, but in this blog, we will focus on understanding higher-order equations, such as cubic and quartic equations.
What Are Higher-Order Equations?
Higher-order equations are mathematical equalities that contain an unknown value, or variable, which is raised to a power greater than two. These include cubic equations (third-degree), quartic equations (fourth-degree), and so on. While in GCSE Maths you'll primarily work with quadratic equations (second-degree), higher-order equations come up in A-level Maths and further education.
Before attempting to solve higher-order equations, it’s essential to understand factorisation and synthetic division, two methods used in the UK curriculum to simplify and solve these types of equations.
- Factorisation: This involves breaking down a complicated polynomial into simpler components, known as factors, that can be solved more easily.
- Synthetic Division: This is a modern method of dividing polynomials, often used to find one solution of a cubic or quartic equation. This method also makes it easier to break the equation into smaller parts.
How to Solve Cubic Equations
Just like with quadratic equations, the first step is to simplify the equation. Although you will usually encounter equations already simplified, it’s essential to group the factors together until the structure looks something like this:
x³ - 2x² - x + 2 = 0
Once you have an equation in this form, follow these steps to solve it:
- Divide the values: Start by using synthetic division to divide the polynomial values by the divisors of the constant (the term without x).
- Example: In the equation x³ - 2x² - x + 2 = 0, you would divide the values "1, -2, -1, and 2" by the divisors of the constant term (in this case, the divisors of 2 are 1 and 2, both positive and negative).
- Test the divisors: Try each divisor until the last number in your division equals zero.
- Example: Dividing "1, -2, -1, and 2" by 1 using synthetic division results in "1, -1, -2, and 0."
- Identify a solution: When the result is zero, you've found a solution. The first solution is the divisor used for the synthetic division. In this case, the first solution is 1.
- Form a quadratic: After obtaining one result, you can form a quadratic equation using the results of the division.
- Example: The new equation is x² - x + 2 = 0.
- Solve the quadratic: Finally, solve this quadratic equation using the quadratic formula to find the remaining solutions.
How to Solve Quartic Equations and Beyond
For equations beyond the third degree, the process is the same. First, simplify the equation, then use synthetic division or factorisation techniques to break the equation down into smaller parts.
For quartic equations, continue dividing using synthetic division until you can simplify it into a quadratic equation, which can then be solved easily.
In cases where you're working with equations of an even higher degree, keep applying these methods until you've broken the equation down to a manageable level. While this may seem daunting, once you've mastered the process, solving these equations becomes much more straightforward.
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Complex maths concepts might make you feel overwhelmed at first, but with the right approach, they can be learned systematically. Always remember, you're not alone, many students experience the same situation. If you’re struggling to understand these concepts on your own, consider finding a private A-level maths tutor who can help you build confidence and improve your understanding. On FindTutors, you can choose to find an in-person or online private tutor, and filter the results to suit your needs.