Solving Quadratic Equations

(1) Solving Basic Quadratic Equations

(2) Graphing Quadratic Equations (Parabolas)

(3) Solving Quadratic Equations Using the Quadratic Formula

(4) Deriving the Quadratic Formula

(5) Quadratic Equation Worksheets

Quadratic Equations - Any single variable equation where the highest power is two. The simplest type of quadratic equation occurs when there is one variable with power two and no other variables.

Procedure - Basic Quadratic Equations

1) Solve the equation so that the x-squared(or whatever variable in the problem) term is by itself on one side of the equation with coefficient one.

2) Take the square root of both sides. On the side with the number be sure to add +/-, because the positive and negative value will solve the equation.

3) Simplify the right side of the equation by simplifying the square root.

Special Case - When you solve for the variable squared if the other side is negative the answer is "no real solution". It is not possible to find a real square root of a negative number.

The following videos show examples of how to apply this procedure.

Just like linear equations it important to look at the graph of quadratic equations. The graph of a quadratic equation is referred to as a parabola and has the general shape of a cross between a "u" and a "v".

There are two specific ways to graph quadratic equations. In this section we will be looking at using a table of values to determine the graph. In order to do this we have to determine what the vertex is. The vertex is the point at the bottom(or top) of the parabola.

We start by finding the "x" coordinate of the vertex. The formula for finding this is x = -b/2a. The a and b come from y = ax2+bx+c. Once the x coordinate of the vertex is determined it is placed on the table of values in such a way that two more values can be placed above it and two below.

The next step is to fill those empty x-values by counting up and down from the x-coordinate of the vertex. Once you have done this you fill in the table of values by plugging the x-values into the original equations and finding the y-values. After you have done this you graph the points and draw in the graph.

The most common method of solving quadratic equations is using the quadratic formula. The quadratic formula is used with the standard form of a quadratic equation which is ax2+bx+c=0. The quadratic formula is:



It is key to first figure out a,b and c. To do this you solve the quadratic equation for zero. The number in front of the x2 is a, the number in front of x is the b and the number that is by itself is c.

The following video examples show how to find the solutions to a quadratic equation using the quadratic formula.

A third way to solve quadratic equations is to solve by factoring. This is based on the Multiplication Propert of Zero.

Multiplication Property of Zero

If a*b = 0 then either a = 0 or b = 0.

Thus we set the quadratic equation equal to zero and factor. Once we have factored the quadratic equation then we set each factor equal to zero. We do this because when one of the two factors is equal to zero then the quadratic equation is equal to zero.

Watch the following videos to see some examples of how to solve quadratic equations by factoring.