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Solving
a quadratic (or any kind of equation)
by factoring it makes use of a principle
known as the zero-product rule.
Zero Product Rule
If
ab = 0 then either a = 0 or b = 0
(or both).
In
other words, if the product of two things is zero then one of those two
things must be zero, because the only way to multiply something and get zero
is to multiply it by zero.
Thus,
if you can factor an expression that is equal to zero, then you can
set each factor equal to zero and solve it for the unknown.
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The expression must
be set equal to zero to use this principle
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You can always make any
equation equal to zero by moving all the terms to one side.
Example:
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Given:
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x2 – x = 6
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Move all terms to one side:
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x2 – x – 6 = 0
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Factor:
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(x – 3)(x + 2) = 0
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Set each factor equal to zero and solve:
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(x – 3) = 0 OR (x + 2) = 0
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Solutions:
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x
= 3 OR x = -2
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If a quadratic equation has no constant term
(i.e. c = 0) then it can easily be solved by factoring out the
common x from the remaining two terms:
Then, using the zero-product rule, you set each
factor equal to zero and solve to get the two solutions:
x = 0 or
ax + b = 0
WARNING:
Do not divide out the common factor of x or you will lose the x = 0
solution. Keep all the factors and use the zero-product rule to get the
solutions.
When a quadratic has all three terms, you can still solve it with the
zero-product rule if you are able to factor the trinomial.
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Remember, not all trinomial
quadratics can be factored with integer constants
If it can be factored, then it can be written as a product of two binomials.
The zero-product rule can then be used to set each of these factors equal to
zero, resulting in two equations that are both simple linear equations that can
be solved for x. See the above example for the zero-product rule to see
how this works.
A more thorough discussion of factoring trinomials may be found in the
chapter on polynomials, but here is a quick review:
1.
Clear fractions (by multiplying through by the common
denominator)
2.
Remove common factors if possible
3.
If the coefficient of the x2 term is
1, then
x2 + bx + c = (x + n)(x + m),
where n and m
i.
Multiply to give c
ii.
Add to give b
4.
If the coefficient of the x2
term is not 1, then use either
a. Guess-and Check
i.
List the factors of the coefficient
of the x2 term
ii.
List the factors of the
constant term
iii.
Test all the possible
binomials you can make from these factors
b. Factoring by Grouping
i.
Find the product ac
ii.
Find two factors of ac that add to give b
iii.
Split the middle term into the sum of two terms, using
these two factors
iv.
Group the terms into pairs
v.
Factor out the common binomial
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