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 4.5d:

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 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 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 :

Subtract from both sides of the equation:

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.

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 0.12530322634 and one at -0.1535874976.

The answers are 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=0.12530322634 by substituting 0.125303226341 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 -0.02354671551 for x, then x=-0.02354671551 is a solution.

Check the solution x=-0.1535874976 by substituting -0.1535874976 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 -0.1535874976 for x, then x=-0.1535874976 is a solution.

The solutions to the equation are 0.12530322634 and - 0.1535874976.

If you are still not sure of this material and would like to start over, click on Equation.

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Author: Nancy Marcus