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ALG2 Unit 3 – Solving Quadratic Equations

Solving quadratic equations is different than solving linear equations.  When solving linear equations we are looking for the value of x that makes the equation true.  When solving quadratic equations we are looking for the values of x that make the equation equal to zero.  If we explore a quadratic we notice that the shape of the graph(parabola) can have two different values when x=o, one value or none at all.  When x=0, that represents the x-intercept of the graph.  What also separates quadratic equations from linear equations is that when solving there is often two variables.  One to the second power and one to the first.  To solve problems like this we cannot simply isolate the variable because there are two of them.  We cannot combine them like linear equations because they are not “like terms”.

So then what do we do?

We factor and apply the zero product property.  Factoring allows us to rewrite our quadratic as the product of two factors.  When we have the product of two factors equal to zero, the zero product property tells us that one or both of those factors has to equal zero for the equation to be true.  So we set both factors to zero and solve for x.  Those will be our solutions to our quadratic equation because they make the equation equal to zero.

Just to note not always will our quadratics have two variables.  For an equation to be a quadratic the only stipulation is that one variable is raised to the second power.  If you have only one variable raised to the second power and no other variables.  You can use inverse operations and take the square root last to undo the power.  This is called the square root method.  I will show this method as well as what to do with only two terms of a quadratic in this first video.

Solving a quadratic when there are only two terms

Solving quadratics with three terms takes a different factoring approach. I will start of showing you step by step how to factor and progress into factoring in your head

Solving a quadratic with three terms when a=1

In this next video, we will explore how to factor when a is not 1. We can do this the long way or the fast way. However to factor polynomials like this in your head you need experience and practice. In this video I will show both ways but it will be up to you to gain your experience.

Solving a quadratic with three terms when a is not 1

While these techniques are great. There are some shortcuts if you can find them. In this video I will show you how to bypass factoring by using special factoring techniques such as difference of two squares and perfect square trinomials

Solving a quadratic using difference of two squares and perfect square trinomials

Not always will problems will be factorable. For solving that is not an issue. For all problems that are non-factorable we have two options. Completing the square and quadratic formula. Which method you prefer is up to you. While you should know completing the square, the quadratic formula more adaptive to harder problems.

Solving a quadratic using completing the square

Solving a quadratic using the quadratic formula


 

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