## 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
4. Graphing

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

Solve for x in the following equation.

Problem 4.5c:

are the exact answers using the Completing the Square Method.

even though the two answers look different, they are equivalent because both yield the same approximate answers of and

Solution:

Simplify the equation .

Divide both sides by

Method 1: Factoring

The equation is not easily factored. Therefore, we will not use this method.

Method 2: Completing the square

Add to 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 :

Subtract from both sides of the equation :

are the exact answers are approximate answers.

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 1 for a, for b , and for c in the quadratic formula and simplify.

are the exact answers are approximate answers.

Method 4: Graphing

Graph the equation, (the left side of the original equation). Graph (the x-axis). 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, one at 6.57797305561 and one at -10.31963044.

The answers are 6.57797305561 and -10.31963044. 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 x = 6.57797305561 by substituting 6.57797305561 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 6.57797305561 for x, then x = 6.57797305561 is a solution.

Check the solution x = -10.31963044 by substituting -10.31963044 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 -10.31963044 for x, then x = - 10.31963044 is a solution.

The solutions to the equation are 6.57797305561 and -10.31963044.

If you would like to review the solution to problem 4.5d, click on Problem

If you would like to go back to the beginning of the quadratic section, click on Quadratic

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

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Author: Nancy Marcus