Introduction to Quadratic Equations

So far in Algebra 1 you will have only dealt with equations that use variables to the first power. That is, you haven’t had to deal with any equations that use variables that have an exponent. That’s about to change as you start to deal with quadratic equations, which involve polynomials of just one variable with the variable raised to the second power. The following are some examples of quadratic polynomials:

x^2 + 5x + 4
4y^2 - 9
1 - a - a^2

All of these polynomials have only one variable, and their variable is raised to the second power. That’s what makes them quadratic.

When we dealt with linear equations, we had a basic form for writing out the equation. That form was y = mx + b. For quadratics, we have standard form as well, and it looks like this:

ax^2 + bx + c = 0

Before you do anything with a quadratic equation, you’re going to need to put it into this form. Here we’re going to look at an example of putting a quadratic equation into the standard form for practice and to show you what the process looks like.

3x^2 + 4x - 5 = 4x^2 - 2x + 1

Let’s move everything to the left side of the equation.

3x^2 + 4x - 5 = 4x^2 - 2x + 1
3x^2 - 4x^2 + 4x - 5 = -2x + 1
-x^2 + 4x - 5 = -2x + 1
-x^2 + 4x + 2x - 5 = 1
-x^2 + 6x - 5 = 1
-x^2 + 6x - 5 - 1 = 0
-x^2 + 6x - 6 = 0

And now we have it in the regular form.

When we graph quadratic equations, we’re going to get a shape that has a bit of a curve to it. This shape is called a parabola.

A neat feature of quadratic equations is that they can have zero, one, or two real solutions. For example, here are three equations:

x^2 - 6x + 9 = 0
y^2 + 4x + 3 = 0
z^2 + 1 = 0

In the first equation, x = 3 is the only solution. In the second equation, y = -1 and y = -3 are both solutions. In the final equation, there are no real solutions.

Factoring Using the GCF – Using the greatest common factor allows us to factor a lot of equations quickly and easily.

Factoring Trinomials as a Product of Binomials – When two binomials are multiplied together to get a trinomial, a special factoring process is needed.

Solving Quadratics with Factoring – Solving quadratic equations is one of the most important topics in Algebra 1.

Imaginary Unit and Complex Numbers – The real numbers are a subset of the complex numbers, which includes values like the square roots of negative numbers.

Quadratic Formula – The only algebraic method to guarantee that you can solve a quadratic equation by hand.

Patterns When Factoring – There are some factoring patterns that are worth knowing because of the time they can save you.

Completing the Square – An algebraic tool that’s needed to be able to graph quadratics.

Graphing Quadratics – Learn to graph quadratics by hand without having to plot a series of points.