A quadratic can be a monomial, binomial or trinomial where the the highest degree is two. For example x^2,3x^2-5, 2x^2+3x, and (1/2)x^2 +4x-2 are all examples of quadratics. The two different examples we will discover are an quadratic equation and function. The only difference between then two within the context of how we will use them is their output variables.

Factoring is the process of rewriting an expression as a product of it's factors. We start by learning how to determine the factors of a number. For instance the factors of 12 are 12,6,4,3,2,1 because they evenly divide into 12. If we were going to write 12 as a product of it's factors one example would be 6*2. This is our goal for factoring expressions with variables as well. We can always check our answer by multiplying the factors together. Factoring is important because it allows us to transform an expression where we can apply the zero product property for solving and simplify the expression when needed.

When factoring out a GCF from an expression we need to know what a GCF really means. "GCF" mean greatest common factor. So what we are looking for is a common factor meaning a number or term that evenly divides into each term. When we say factor out we really mean divide out but rewrite the expression as a product.

Solving a quadratic means finding the values of x or your input variable when your output variable (y) is zero. This is the same for functions. When the output variable is zero then our point lies on the x-axis of the graph so when we think of the graphical solution it is the x-intercepts of the quadratic.

The zero product property is essential to solving quadratics by factoring. What the zero product simply states is that if you have the product of two terms or expressions that equal zero then one of those solutions has to equal zero. Therefore to find the value of each term or expression we set them equal to zero and solve.

There are a couple of ways to factor a quadratic when a=1. Throughout this course I will talk about a couple of them. While the goal is to eventually factor in your head. At times there will be problems that are hard to factor or not even possible. Therefore we will use a couple of techniques to ensure our answer is correct.

There are multiple ways to solve a quadratic when a is not equal to one. In this video we will discuss how to do that mentally as well as two other approaches where you can break down the problem to ensure your answer is correct and it can be factored.

To solve a quadratic by applying the difference of two squares it is very important that we have our quadratic set equal to zero. We will then apply the factoring technique of the difference of two squares and use the zero product property to solve.

Process for solving by factoring a perfect square trinomial:

1.) Set your equation equal to zero

2.) Factor GCF if necessary to obtain square terms

3.) Determine if your coefficient and constant are square numbers and can be written as factors by taking the square root.

4.) Write your perfect square as a binomial squared to solve using the square root method or write as the product of two binomials and solve using the zero product property.

Process for solving by square root method:

1. Set your equation equal to zero
2. Undo addition and subtraction first from the variable
3. Undo multiplication and division second from the variable
4. Undo squaring by taking the square root of both sides. Be sure to include the positive and negative values.
5. Taking the square root of a negative number produces no real solutions, or two imaginary solutions.

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Here are the general steps for solving a quadratic by completing the square

1.) Set your equation equal to zero

2.) Factor if necessary out of your quadratic and linear terms so a=1

3.) (b/2)^2 then add to your binomial to make it a trinomial and add to the other side of the equation.

4.) Factor your perfect square trinomial to a binomial squared.

5.) Apply inverse operations by adding or subtracting on both sides to eliminate your c

6.) Take the square root on both sides to eliminate the square

7.) Add or subtract your factor to isolate your variable

To solve a quadratic we will want to understand the discriminant and determine it's value. This will one help us solve using the quadratic formula but also give us an idea of the roots of the equation. The discriminant is b^2-4ac which is under the square root in the quadratic formula. After we have determined it's value and noted the number of roots we can simplify the remaining equation to solve for x.