SOLVING QUADRATIC EQUATIONSSOLVING QUADRATIC EQUATIONS
Note:
Factoring
Completing the Square
Quadratic Formula
Graphing
Solve for x in the following equation.
Problem 4.5a:
Answer:
are the
exact answers 
are approximate
answers.
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:
are the exact answers 
are approximate
answers.
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.
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 
for a, 
for b, and 
for c in the quadratic formula and simplify.
are the exact
answers are approximate
answers.
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.
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.
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
If you would like to go back to the equation table of contents, click on
Contents
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