The Ultimate Guide to Algebra Formulas with Examples

If there is one thing that comes to mind when people think of algebra and mathematics, it is the formulas. As a branch of mathematics, the study of algebra involves many basic formulas and many advanced formulas in order to make sense of the patterns that we see in real life.

In this guide, I want to provide you with 25 algebra formulas with examples. My goal is for you to develop an understanding of the basics of algebra, so that you can start to solve any complex problems that you come across!

Whether you are looking to learn more about the quadratic formula or dig deeper into logarithm laws, this guide has you covered!

What is Algebra?

Algebra is a branch of maths that involves mathematical symbols and numbers that work together to form a mathematical expression and mathematical equations. Algebra also focuses on the rules used to rearrange algebraic expressions, solve equations, and represent relationships between real life variables.

As students advance in their studies of algebra, they start to notice letters making their way into their math courses. While this may sound confusing, letters are used as variables to represent unknown quantities. In algebra, we use these variables and mathematical operations such as addition, subtraction, multiplication, and division to construct algebraic expressions and algebraic equations.

In algebra, you will often find yourself attempting to determine the values of unknown variables using equations, specific rules, and mathematical operations. One very important rule you might know about already is that what is done to one side must be done to the other.

Algebra equations are like scales: if you change one side of the scale, it becomes out of balance. In order to balance it, you must make the same change on the other side of the scale.

Since algebra is a fundamental area of mathematics that forms the basis for many other mathematical and scientific disciplines, the importance of algebra cannot be overstated!

What are Algebra Maths Formulas?

An algebraic equation or a “math formula” is a way of summarizing a mathematical relationship. Math formulas contain a combination of unknown values (the unknown variable x, for example), numbers, and typically, an equal sign.

While x and y tend to be the most commonly used letters, the variables that make up an algebraic equation can be any letters or mathematical symbols.

Math formulas allow us to place different values into the subject of the formula in order to see what happens. For example, if you have a formula for the surface area of a cone, you could sub in different values for the radius in order to see how the surface area changes with each.

Mathematical formulas are very useful because they allow us to make predictions based on patterns that exist. This makes algebra formulas very applicable to a variety of real life applications.

For example, it is possible to use math formulas to predict how much money you will have in the future if you invest $1500 into a high interest savings account right now. A person studying social science might create a mathematical formula that predicts population growth over time. The possibilities are truly endless!

25 Algebra Formulas with Examples

The formulas of algebra range from using math basic math operations to much more complex operations. This list of algebra formulas with examples will provide you with a variety of levels of difficulty.

Let’s begin by considering one of the more basic algebraic formulas that you will encounter in your studies of elementary algebra!

The Distributive Property

One of the most common rules that you will apply in algebraic problems is the distributive property (sometimes referred to as the distributive axiom). This property states that $z(x + y) = xz + yz$ and $z(x – y) = xz – yz$. The final result will always be a simpler expression without brackets!

The distributive property is a common step that is taken to simplify algebraic expressions in a wide variety of mathematical applications and problems.

Below is an example of the distributive property in action!

Algebraic Identities

There are a collection of algebraic identities that can be used to expand and simplify a product of binomials. Each algebraic identity shown in the below formulas can be explained using the Binomial Theorem.

The following algebraic identities involve trinomials that are perfect squares:

Square of the Sum: $(a + b)^2 = a^2 + 2ab + b^2$
Square of the Difference: $(a – b)^2 = a^2 – 2ab + b^2$
Difference of Squares: $a^2 – b^2= (a + b)(a – b)$

This quick example of the square of the sum formula, will help you see how this formula works in practice.

The following formulas are useful when expanding and simplifying binomials. These are especially helpful when working with a cubic equation, as these tend to be tricky to work with.

Product of Two Binomials: $(x + a)(x + b) = x^2 + (a + b) x + ab$
Cube of a Binomial (Sum): $(a + b)^3 = a^3 + b^3 + 3ab (a + b)$
Cube of a Binomial (Difference): $(a – b)^3 = a^3 – b^3 – 3ab (a – b)$

General Form of The Quadratic Equation

The general form of the quadratic equation (also known as standard form) is the equation $f(x) = ax^2+bx+c$. Notice that the power on x decreases on each successive term until the constant term of c. This constant value tells us the y-intercept of the quadratic function. The general form of the quadratic equation is often considered to be the most simplified form.

The Quadratic Formula

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

The above formula is the famous quadratic formula! While it may look complex, the quadratic formula is considered a basic algebra formula. This powerful formula allows us to determine the x-intercepts of any quadratic function (as long as it has them!).

While the quadratic formula is helpful, many students (and teachers) prefer factoring trinomial expressions to solve quadratic equations in order to find their x-intercepts as it is a much faster process. Others prefer to rely on the quadratic formula as a way of avoiding factoring altogether (however, I strongly discourage this as factoring is a very useful skill that helps us write expressions in different forms!).

The values of the quadratic formula correspond to the values of a, b, and c in a quadratic trinomial.

Pythagorean Theorem

Measurement is a branch of maths that sees many applications of algebra in the real-world. I want to share a few algebra formulas with examples here that relate to measurement in the real world. One of the most famous is the Pythagorean Theorem.

$$a^2+b^2=c^2$$

This formula tells us that the square of each leg of a right triangle is equal to the sum of the hypotenuse. It is important to note that the hypotenuse is isolated on the right-hand side in the above equation. While the values of a and b are interchangeable, the hypotenuse is not!

For a quick example, of the Pythagorean Theorem in action, check out this short example video from my YouTube channel!

Trigonometric Ratios

Arguably some of the most important algebra formulas in mathematics, the trigonometric ratios are another set of useful algebra formulas that involve right triangles. These mathematical equations relate the side lengths of a right triangle to its angles.

$sin \theta = \frac{o}{h}$, where o is the side opposite to the angle, and h is the hypotenuse.
$cos \theta = \frac{a}{h}$, where a is the side adjacent to the angle, and h is the hypotenuse.
$tan \theta = \frac{o}{a}$, where o is the side opposite to the angle, and a is the side adjacent to the angle.

These ratios can be used to solve complex problems in trigonometry, a very important topic in studies of algebra and mathematics. For example, a common application is when solving angle of elevation and depression problems.

angle of depression of 30 degrees from the top of a building to a car

Surface Area

When it comes to mathematical applications of algebra formulas with examples, surface area is an easy one to connect to real life. Surface area is a measurement of the amount of space that covers a three dimensional object. Mathematical problems involving surface area are everywhere, and we need a different formula for every shape that we encounter!

For example:

Surface Area of a Rectangular Prism: $SA=2(lh +wh + lw)$, where l is the length, w is the width, and h is the height.
Surface Area of a Cylinder: $SA=2 \pi rh + 2 \pi r^2$, where r is the radius, and h is the height.
Surface Area of a Cone: $SA = \pi r^2 + \pi r s$, where r is the radius, and s is the slant height.
Surface Area of a Sphere: $SA = 4\pi r^2$, where r is the radius.

Exponential Laws

The exponential laws are a useful set of rules for simplifying exponential expressions that contain different powers. Note that these laws do not work for expressions with different bases.

Some common laws of exponents are:

Multiplying powers, add the exponents: $x^a \times x^b = x^{a + b}$
Dividing powers, subtract the exponents: $x^a \div x^b = x^{a – b}$
Power of a power: $(x^a)^b = x^{ab}$
Negative exponent rule: $x^{-a} = \frac{1}{x^a}$

Early in my teaching career, I wrote and performed a cheesy exponent law song to help my class remember these laws. Check it out!

Important Log Laws

Working with expressions in logarithmic form is a valuable skill in the study of algebra. The important log laws are similar to the exponential laws, but have their own unique set of rules that you must follow.

Product rule for logarithms: $log_{a}(x \times y) = log_{a}x + log_{a}y$
Quotient rule for logarithms: $log_{a}(x \div y) = log_{a}x – log_{a}y$
Quotient rule for logarithms: $log_{a}(x^y) = ylog_{a}x$
Base change rule for logarithms with different bases: $log_{b}x = \frac{log_{c}x}{log_{c}b}$

Watch this log laws video for examples of how these important log laws can be used to simplify algebraic expressions.

Using Algebra Formulas With Examples

One important thing I have noticed in my time as a classroom teacher is that just “knowing the formulas” is not enough to guarantee success. Instead, the best thing you can do is to make sure you know how and when to use each formula. This is especially important when solving real-world algebra problems!

Algebra is more than just plugging in numbers into formulas. There is a lot of critical thought that goes into selecting and applying a specific formula over another! When it comes to using algebra formulas, practice is the key to mastery!

I hope you found this guide to algebra formulas with examples helpful as you continue your studies of algebra. While this list does not contain every algebraic formula, it should provide you with a good overview of just what is possible in the study of algebra.

Did you find this guide to algebra formulas with examples helpful? Share this post and subscribe to Math By The Pixel on YouTube for more helpful mathematics content!