SOLVING QUADRATIC EQUATIONS

Note:

• A quadratic equation is a polynomial equation of degree 2.

• The ''U'' shaped graph of a quadratic is called a parabola.

• A quadratic equation has two solutions. Either two distinct real solutions, one double real solution or two imaginary solutions.

• There are several methods you can use to solve a quadratic equation:
  1. Factoring
  2. Completing the Square
  3. Quadratic Formula
  4. Graphing

• All methods start with setting the equation equal to zero.

Solve for x in the following equation.

Example 1:

Set the equation equal to zero by subtracting 3 x and 7 from both sides of the equation.

Method 1: Factoring

The left side of the equation is not easily factored, so we will not use this method.

Method 2: Completing the Square

Subtract 15 from both sides of the equation .

Add to both sides of the equation:

Factor the left side and simplify the right side :

Take the square root of both sides of the equation:

Add to both sides of the equation:

and

Method 3: Quadratic Formula

The quadratic formula is

In the equation ,a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -3 for b, and 15 for c in the quadratic formula and simplify.

and

Method 4: Graphing

Graph y= the left side of the equation or and graph y= the right side of the equation or y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.

You can see from the graph that there are no x-intercepts. This means that there are no real answers; the solution are two imaginary numbers.

The answers are and These answers may or may not be solutions to the original equations. You must verify that these answers are solutions.

Check these answers in the original equation.

Check the solution by substituting in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

  
  

  
  

  
  

  
  

  
  

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value for x, then is a solution.

Check the solution by substituting in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

  
  

  
  

  
  

  
  

  
  

• Right Side:

  
  

  
  

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value for x, then is a solution.

The solutions to the equation are and

Comment: You can use the solutions to factor the original equation.

For example, since ,then

Since , then

Since the product

then we can say that

This means that and are factors of

If you would like to work another example, click on Example.

If you would like to test yourself by working some problems similar to this example, click on Problem.

If you would like to go back to the equation table of contents, click on Contents.

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