SOLVING QUADRATIC EQUATIONSSOLVING QUADRATIC EQUATIONS
Note:
Factoring
Completing the Square
Quadratic Formula
Graphing
Solve for x in the following equation.
Example 3:
The equation is already set to zero.
Almost all precalculus students hate fractions before they renew their love
for them. The following steps will transform the equation into an equivalent
equation without fractions.
If you have forgotten how to manipulate fractions, click on Fractions for a review.
If you have forgotten what equivalent means, think of a dollar. You can
represent the dollar with a dollar bill, 10 dimes, 20 nickels, or 100
pennies. All of these are equivalent because all have the value of a
dollar. Got it. If not, click on Equivalence 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 56. All the denominators 8,
28, and 7 divide into 56 evenly.
Method 1:Factoring
The equation 
can be factored as follows:
Method 2:Completing the square
Divide both sides of the equation 
by 21.
Add 
to both sides of the equation: 
Simplify: 
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:
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.
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 
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 
by substituting 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 for x,
then is a solution.
Check the solution 
by substituting 
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.
for x,
then is a solution.
The solutions to the equation 
are and
Comment:Recall that when we solved this equation by factoring, we factored
the expression not the original expression
The product of the
factors of 
does not equal the original expression because the product of the first terms of the factors must equal the first term of the original expression.
We need to add a constant factor.
Product of factors 
equals 
What number do I multiply 21 by to get 
Let's us see if 
equals the original
The factors of 
are
We have illustrated that the solutions to the equation are
and
If you want to verify that you know how to work this type of problem by
testing yourself over problems similar to the one above, click on Problem
If you would like to go back to the equation table of contents, click
on Contents
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