SOLVING QUADRATIC
EQUATIONS SOLVING QUADRATIC
EQUATIONS 
Note:
Factoring
Completing the Square
Quadratic Formula
Graphing
Solve for x in the following equation.
Example 4:
The equation is already set to zero.
If you have forgotten how to manipulate fractions, click on Fractions for a review.
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 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 462. All the denominators
6, 231, and 77 divide into 462 evenly.
How do we know this? 
and 
If we choose the set of factors
the set of factors of each denominator can be found in this set. This means
that each of the denominators divides evenly into the product 462.
Method 1:Factoring
The left side of the equation 
can be factored
as follows :
The answers are 
Method 2:Completing the square
Divide both sides of the equation 
by 77
.
Simplify: 
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:
Add 
to both sides of the equation:
The answers are
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.
The answers are
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.
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.
Original Equation : 
Product of factors 
equals 
What number do I multiply 77 by to
get 
Let's us see if 
equals the original We can do this using equivalence (evaluating both
expressions by the same number). If the answers are the same, you have
factored the original expression correctly. We can also this by
algebraically by multiply the factors out and seeing if they equal the
original expression
Equivalence:
Evaluate both expressions at x=2
This confirms the factors are correct using equivalence
Algebraically
Since the original expression is , we have shown the factors are correct algebraically.
We have illustrated that the solutions to the equation are
and
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