Common Algebraic Equations: Linear, Quadratic, Polynomial, and More

Watch this video lesson to see the kinds of equations that you will come across most often in algebra. Learn to distinguish them by just looking for the identifying components of each equation.

Common Algebraic Equations

In algebra, there are some equation types that you will come across more often than others. You will find that if you can identify the type of equation that you are working with, then it becomes easier to work with the problem since you will know the properties of the equation. In this lesson we will cover six common algebraic equations.

Linear

The first one is called the linear equation. The general form of these equations is y = mx + b, where m and b are numbers and m cannot be zero. The way to identify these types of equations is to look for an x with no exponents. The x should be the only variable you see other than the y. You should not have any other exponents or square roots. The x is also always in the numerator, never in the denominator.

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These equations are called ‘linear’ because when you graph them, you end up with a single line. So, to help you remember that you should only see one x, think of linear as having one line, and link the one line to the one x in your head. For example, y = 4x + 3 is a linear equation. Note that you see the x and no other x‘s. We can start building a table to keep all of these equations and their names organized.

Equations table

Quadratic Equation

The second common type of equation is the quadratic equation. This type of equation has a general form of ax^2 + bx + c = 0, where a, b and c are numbers and a is never zero. The other two letters, b and c, can be zero.

The key thing to look for here is the x^2. The exponent of 2 is the highest and you should not see any higher exponents in the equation. If the b is not a zero, then you will also see an x with no exponent. You should not see more x‘s than these two. An example of a quadratic equation is 4x^2 + 3x + 1 = 0. Do you see how the highest exponent is two? We can add this to our table.

equation table

Cubic

The next type is the cubic equation, which has the general form of ax^3 + bx^2 + cx + d = 0, where a, b, c and d are numbers but a cannot be zero. The way to identify these types of equations is to look for the x^3. The 3 should be your highest exponent.

If b and c are not zero, then you will also have an x^2 and an x term, but your terms will never have an exponent higher than 3. For example, x^3 = 0 is an example of a cubic equation. Notice that 3 is the highest exponent here and our b, c and d are zero, but our a is a 1. Adding this information to our table, we get this.

equation table

Polynomial

While your linear, quadratic and cubic equations limited your highest exponent to 1, 2 and 3 respectively, the polynomial equation takes away that limit. A polynomial is of the form:

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where all the a‘s are numbers. The word ‘polynomial’ means ‘consisting of several terms,’ and as you can see, this opens up a whole world of equations that includes linear, quadratic and cubic equations. The only restrictions are that the exponents have to be positive, whole numbers. They cannot be negative, nor can they be fractions. An example of a polynomial is the equation 5x^6 + 3x^2 + 11 = 0. Let’s add this to our table.

equation table

Rational

Now, if you take a polynomial and divide it by another polynomial, you will have your rational equation. You can say that a rational equation is the fraction of two polynomials. An example of a rational equation is the equation (4x^2 + 3) / (x + 5) = 0. We can now add this to our table.

equation table

Radical

Our last common algebraic equation is the radical equation, which is an equation that involves the radical symbol. So an equation with a square root is a radical equation. So is an equation with the third root. If the equation has the radical symbol – the one used for square roots – then the equation is a radical equation. An equation such as 4x + sqrt3 = 0 is an example of a radical equation because it has the radical symbol in the equation. Now we can finish our table by adding this last bit of information.

equation table

Lesson Summary

Now we can review what we’ve learned. We’ve learned that in algebra, there are several common equation types that we will see over and over again. Being able to recognize these makes it easier to problem solve because we will know the type of equation we are working with. We can summarize the types of equations we covered in this table that will be worth copying or memorizing.

Equation General Form Example
Linear y = mx + b y = 4x + 3
Quadratic ax^2 + bx + c = 0 4x^2 + 3x + 1 = 0
Cubic ax^3 + bx^2 + cx + d = 0 x^3 = 0
Polynomial
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5x^6 + 3x^2 + 11 = 0
Rational Polynomial / Polynomial (4x^2 + 3) / (x + 5) = 0
Radical Equation with radical symbol 4x + √3 = 0

Learning Outcome

By the end of this lesson you will be able to identify and discuss the characteristics of linear, quadratic, cubic, polynomial, rational and radical equations.

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