Equations are a pretty integral part of mathematics. In order to understand slightly more complex concepts such as linear and quadratic equations, we first need to know what equations are. In this post, we’re going to go over all these topics, and then some, so keep reading!

What are equations?

In mathematics, an equation is a statement that demonstrates the equality of two expressions. These two expressions are separated only by an equals sign (=). They can consist of any number of values and operations, with the only requirement being the expressions have to be equal.

We usually say an equation has two sides: the right-hand side, and the left-hand side. The right-hand side is generally assumed to be 0. However, this doesn’t always have to be true (we will see some examples later).

Equations are made up of three parts. Those parts are:

  • The constants
  • The coefficients
  • The variables

The given numbers that stand alone are called constants. They are called that because they don’t change. This means that in any version of a certain equation, they always remain the same.

The coefficients stand with the variables, and they serve to amplify them.

The variables are the unknowns of an equation, the missing parts if you will.

What are linear equations?

A linear equation is an equation of the form ax + b = 0, where a and b are constants and x is an unknown variable. You can think of this as a function with one input (x) and two outputs (a and b). The word “linear” refers to the fact that all the variables in the equation are to the power of 1.

Sir William Rowan Hamilton invented the linear equation in 1843. They are a part of linear algebra, which is a very important branch, not just in mathematics, but in other sciences, such as engineering, computing, and many more.

Solving linear equations

Solving a linear equation means finding the value of x that makes the equality of the two expressions true. When it comes to the one-variable forms, there is a very simple formula for calculating x:

x = /frac -{b}{a}

This is if we assume a is not 0. If a is 0, there are two possible outcomes. In the case that b is also 0, x can be any number, and the equality will be true. If b is not 0 however, there is no solution to the equation, and it is dubbed inconsistent.

With this formula, you can even use a standard algebra calculator to solve the equation.

For more information about linear equations, you can check out our Linear Equation Calculator!

Quadratic equations

Quadratic equations are secondary algebraic expressions of the form:

ax^2 + bx + c = 0

The word quadratic is derived from the word “quad”, which means square. In other words, this equation is the “equation of degree 2.” One of the uses of this equation is to describe the time when a racket is launched. Quadratic equation represents an equation that appears in the form ax2 + bx + c = 0. X represents the unknown in this expression, and a, b, and c represent known numbers or coefficients. All x values that can satisfy it are called solutions. This type of formula has two solutions. If there is no real solution, there are two complex solutions. A quadratic equation consistently has two roots if complex roots are incorporated, and a twofold root is meant two.

Solving quadratic equations

To solve quadratic equations, we use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

For more information about the quadratic formula, you can check out our Quadratic Equation Solver!

How to use the equations calculator

The equations calculator has two modes you can use, depending on what you need, the linear equation solver and the quadratic equation solver. So, all you need to do is choose which mode you want, enter the equation, and the calculator will give you the solution.

FAQ

What is an equation?

An equation is a statement that demonstrates the equality of two expressions.

What does linear mean in “linear equation”?

The word “linear” refers to the fact that all the variables in the equation are to the power of 1.

How are quadratic equations solved?

Quadratic equations are solved using the quadratic formula: (-b±√(b²-4ac))/(2a).