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

Solve for x in the following equation.

Problem 4.5a:

Solution:

Simplify the equation by simplifying the radicals.

Method 1:Factoring

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

Method 2:Completing the square

Add 5 to both sides of the equation

Divide both sides by :

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 1 to both sides of the equation:

Even though the exact answers above do not look like the answers at the beginning of the problem they are equivalent because they simplify to the same approximate answers.

In the equation , a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Substitute for a, for b, and for c in the quadratic formula and simplify.

Even though the exact answers above do not look like the answers in the Completing the Square method or the answers at the beginning of the problem, they are equivalent because they simplify to the same 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 and one at .

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 x=3.1296792965 by substituting 3.1296792965 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 3.1296792965 for x, then x=3.1296792965 is a solution.

Check the solution x=-1.1296792965 by substituting -1.1296792965 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 -1.1296792965 for x, then x=-1.1296792965 is a solution.

The solutions to the equation are 3.1296792965 and -1.1296792965.

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

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

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Author: Nancy Marcus