Quadratic Equations; Your Complete Guide

From Solving, graphing and writing the equation of a quadratic you will learn all step by step
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  • Lectures 68
  • Contents Video: 6 hours
    Other: 2 mins
  • Skill Level All Levels
  • Languages English
  • Includes Lifetime access
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    Available on iOS and Android
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About This Course

Published 5/2014 English

Course Description

In this course you will discover how to solve quadratic functions and equations using algebraic processes. We will begin by applying the square root method and then move into solving by factoring, completing the square, and quadratic formula. These processes will become very important to understand how to determine solutions of a quadratic as well as reason real world applications. The course will start with identifying essential questions that will provide a theme of what you are expected to learn throughout the course. We will then discover the essential definitions and processes that will be necessary to complete this course.

You will then have the opportunity to try on your own over 100 example problems for each process so that you can master the skills set out in this course. If at anytime you want to check your answer or get stuck. Each problem is accompianed with a step by step tutorial video to help you along.

After you have completed the work you can answer the essential questions at the end of this course or work on the challenge problems to round off your learning and prepare you for the next level.

What are the requirements?

  • Pre-algebra
  • Basic arithmetic skills

What am I going to get from this course?

  • Solve quadratics by factoring
  • Solve quadratics without factoring
  • Writing the Equation of a parabola
  • Graphing Quadratics in Standard form
  • Graphing Quadratics in Vertex form
  • Determine the vertex and axis of symmetry
  • Determine the domain and range
  • Bonus: Complex Numbers

What is the target audience?

  • Algebra 1 students
  • Algebra 2 students
  • College Algebra students
  • Pre-Calculus students

What you get with this course?

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30 day money back guarantee.

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Lifetime access.

Learn on the go.
Desktop, iOS and Android.

Get rewarded.
Certificate of completion.


Section 1: Introduction

In this video I will go over what is needed in this course for you to learn how to algebraically solve a quadratic equation and function. There are many processes that we will use in this course and it is important that you have the fundamentals before you start.

Section 2: Basics for solving quadratic equations with factoring

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.

Section 3: Solving Quadratic Equations by factoring
Overview for how to solve a quadratic by factoring out the GCF
Examples for solving a quadratic by factoring out the GCF
Quiz - Solving a quadratic by factoring out the GCF

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.

Examples for solving a quadratic when a is 1
Quiz - Solving by factoring when a is 1

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.

Examples for solving a quadratic when a is not 1
Quiz - Solving a quadratic when a is not 1

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.

Quiz - Solving a quadratic by using the difference of two squares

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.

Quiz - Solving a quadratic as a perfect square trinomial
Examples for solving a quadratic using special factoring techniques
Section 4: Basics for solving Quadratic Equations without factoring
What is the quadratic formula?
What are the different types of solutions to a quadratic?
Section 5: Solving Quadratic Equations without factoring

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

Examples for solving a quadratic by completing the square
Quiz - Solving quadratics by completing the square

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.

Examples for solving a quadratic using the quadratic formula
Examples for solving a quadratic using the quadratic formula; complex solutions
Quiz - Solving a quadratic using the quadratic formula
Section 6: Writing the equation of a parabola
Examples for completing the square to convert from standard to vertex form
Quiz - Writing the equation of the parabola from standard form to vertex form
Examples for writing the patabola of a quadratic given vertex through a point
Quiz - Writing the equation of a parabola given the vertex and a point
Examples for writing the equation of a parabola given three points
Quiz - Writing the equation of a parabola given three points
Section 7: Basics for graphing quadratics
What is standard form of a quadratic
What is vertex form of a quadratic
What is the vertex?
What is the axis of symmetry?
What is the max and min of a quadratic?
What is the line of symmetry of a quadratic?
Section 8: Graphing Quadratic Equations
How to find the vertex of a quadratic in standard form
Examples for graphing a quadratic using the axis of symmetry and vertex
Examples for graphing a quadratic by converting from standard form to vertex
Examples for identifying the vertex, domain and range in vertex form
Graphing a quadratic in vertex form using transformations
Quiz - Graphing a Quadratic in standard form
Section 9: BONUS - Basics for Complex Numbers
What is a complex number?
What is an imaginary number?
What is the imaginary unit i
Section 10: BONUS - Complex Numbers
Overview for how we add and subtract complex numbers
Examples for adding and subtracting complex numbers
Quiz Adding and Subtracting Complex Numbers
Overview for how we multiple complex numbers
Examples for multiplying complex numbers
Quiz Multiply Complex Numbers
Simplifying expressions with complex numbers using multiple operations
Simplify Expression with multiple operations
Examples for writing an expression using complex numbers the simplifying
Quiz Simplify expression using the imaginary unit i
Overview for how we divide complex numbers
Examples for dividing complex numbers
Quiz Dividing complex numbers
Simplifying complex numbers to a higher power
Quiz Simplifying Complex Numbers to a higher power

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Instructor Biography

Brian McLogan, Current Active High School Math Teacher

I am a high school that is on a mission to improve math education. I was that student that sat in the back of class frustrated with the boredom of class and the lack of understanding. I made the decision to become a math teacher to make a difference in others lives. I knew that with the struggles I had I could relate well to students that struggled with math. With a weak math background I set out to get a degree in mathematics. In was a difficult journey and I worked very hard not just to pass my math classes but to have an understanding of what I was doing. I learned a lot about myself, mathematics and what it takes to be successful in class through my time at college. I want to pass along my experience to you the student so that may have your own success with mathematics.

Instructor Biography

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