Solving equations

Solving equations using algebra is a fundamental aspect of many mathematical disciplines. As such, it is important that students build a strong foundation by learning and practicing the various methodologies used to solve equations.

What is an equation?

In order to solve equations, it is important to understand the concept of an equation. An equation, in its simplest form, is an expression of equality. Equations separate two equal expressions with an equal sign such that we know that the expression on the left side of the equal sign must equal the expression on the right side of the equal sign for the equation to be true. For example:

1 + 2 + 3 = 6

The above is an example of a simple equation. In algebra, equations usually involve variables:

x + 2 + 3 = 6

From the above example, we know that x is 1, but if we didn't, we would have to solve the equation by understanding that both sides are equal. 1 is therefore a solution to the above equation, since it makes the equation true. If we were to substitute any other value for x, the equation would be false.

Solving equations

There is no one method for solving an equation. Because equations come in so many different forms, what works for solving one equation may not work for another; it can be as simple as subtracting one number, or it may involve the use of many different techniques like factoring, expanding, completing the square, using identities, substitution, patterns, and more.

Very generally, to solve an equation, we want to isolate the variable. Isolating the variabler means getting it on one side of the equation such that it is equal to an expression on the other side of the equation, making the expression the solution (e.g. x = solution). Using the simple example above:

x + 2 + 3 = 6

x + 5 = 6

x + 5 - 5 = 6 - 5

x = 1

In the above example, we isolated x by first combining like terms (2 + 3), then subtracting 5 from both sides, leaving us with the expected solution of x = 1. Depending on the equation, it is possible to have multiple/many solutions. This particular example only required the use of addition and subtraction, but as mentioned, there are many different ways to manipulate an equation to facilitate solving it. There are also certain formulas or techniques that can be used to solve specific types of equations.

Quadratic formula

The quadratic formula is a formula that can be used to solve a common and specific type of equation: quadratic equations. Given a quadratic equation in general form (ax² + bx + c = 0), using the quadratic formula just requires plugging in the appropriate values of a, b, and c. Doing so will yield the solutions to the equation. The quadratic formula is as follows:

Quadratic formula:

Refer to their respective pages for more information on the quadratic formula and quadratic equations. Completing the square is another way to solve quadratic equations, and generally involves creating a perfect square trinomial from a quadratic equation.