SOLVING QUADRATIC EQUATIONS
SOLVING QUADRATIC EQUATIONS 
Note:
Factoring
Completing the Square
Quadratic Formula
Graphing
Solve for x in the following equation.
Problem 4.5c: 
Answer: 
are the exact answers using the Quadratic Formula.
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.
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.
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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