SOLVING QUADRATIC EQUATIONSSOLVING QUADRATIC EQUATIONS
Note:
Factoring
Completing the Square
Quadratic Formula
Graphing
Solve for x in the following equation.
Problem 4.4a:
Answer:
are the exact answers 
are approximate answers.
Solution:
Set the equation equal to zero by subtracting 2 from both sides.
If you have forgotten how to manipulate fractions, click on Fractions for a review.
Remove all the fractions by writing the equation in an equivalent form
without fractional coefficients. In this problem, you can do it by
multiplying both sides of the equation by 3.
Method 1:Factoring
The equation 
is not easily factored. Therefore,
we will not use this method.
Method 2:Completing the square
Add 36 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:
Add 6 to 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 
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, 
(formed by
subtracting the right side of the equation from the left side of the
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 14.485281 and one at -2.485281.
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=14.485281 by substituting 14.485281 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 14.485281 for x, then
x=14.485281 is a solution.
Check the solution x=-2.485281 by substituting -2.485281 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 
and 
Comment:You can use the exact solutions to factor the left side of the
original equation minus the right side of the original equation:
Since 
:
Since 
:
The product 
Since 
and 
then we could say
However the product of the first
terms of the factors does not equal 
Multiply 
by 
Let 
s check to see if 
The factors of 
are 
and 
If you would like to test yourself by working some problems similar to this
example, click on Problem
If you would like to go back to the equation table of contents, click
on Contents
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