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 4 :

Rewrite the equation in an equivalent form without denominators by multiplying both sides by 6.

Method 1:Factoring

Since the equation is not easily factored, we will skip this method.

Method 2:Completing the square

Add 5 to both sides of the equation

Divide both sides by 4 :

Add 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.

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. Substitute 4 for a, -90 for b , and -5 for c in the quadratic formula and simplify.

Method 4:Graphing

Graph and 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 two x-intercepts located at 22.5554190546 and -0.055419054595. This means that there are two real answers: x=22.5554190546 and

The answers are 22.5554190546 and -0.055419054595. These answers may or may not be solutions to the original equation. You must check the answers with the original equation.

Check these answers in the original equation.

Check the solution x=22.5554190546 by substituting 22.5554190546 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 x=22.5554190546 for x, then x=22.5554190546 is a solution.

Check the solution x=-0.055419054595 by substituting -0.055419054595 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 -0.055419054595 for x, then x=-0.055419054595 is a solution.

The solutions to the equation are 22.5554190546 and - 0.055419054595.

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

For example, since , then

Since , then

Since the product

Then and are factors of .

But not the only factors.

If we multiply both sides by , we will get :

Therefore is factored as

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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