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.

• The 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 5:

Set the equation equal to zero by subtracting and adding to sides of the equation.

Method 1: Factoring

The expression cannot easily be factored, so we will not use this method.

Method 2: Completing the square

Divide both sides of the equation by 24.

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 :

Subtract from 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 24 for a, 33 for b, and -136 for c in the quadratic formula and simplify.

and

Method 4:Graphing

Graph . The equation represents the left side of the original equation minus the right side of the original equation. The right side of the equation is now zero. 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 two x-intercepts located at -3.165266 and 1.790266. This means that there are two real answers: x = - 3.165266 and 1.790266 .

The answers are -3.165266 and 1.790266.

These answers may or may not be the solutions to the original equation. Check these answers in the original equation.

Check the solution x = - 3.165266 by substituting - 3.165266 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 - 3.165266 for x, then x = - 3.165266 is a solution.

Check the solution x = 1.790260 by substituting 1.790260 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 1.790260 for x, then x=1.790260 is a solution.

The solutions to the equation are -3.165266 and 1.790260.

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 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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