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The solutions to a quadratic
equation can be found directly
from the quadratic formula.
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The
equation
ax2
+ bx + c
= 0
has
solutions
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The advantage of using the
formula is that it always
works. The disadvantage is
that it can be more time-consuming
than some of the methods previously
discussed. As a general rule
you should look at a quadratic
and see if it can be solved
by taking square roots; if
not, then if it can be easily
factored; and finally use
the quadratic formula if there
is no easier way.
·
Notice the plus-or-minus
symbol (±) in the formula. This is how you get the
two different solutions—one
using the plus sign, and one
with the minus.
·
Make
sure the equation is written
in standard form before reading
off a, b, and
c.
·
Most
importantly, make sure the
quadratic expression is equal
to zero.
The formula requires you
to take the square root of
the expression b2
– 4ac, which is called
the discriminant because
it determines the nature of
the solutions. For example,
you can’t take the square
root of a negative number,
so if the discriminant is
negative then there are no
solutions.
| If b2 – 4ac > 0 |
There are two distinct real
roots |
| If b2 – 4ac = 0
|
There is one real root |
| If b2 – 4ac < 0
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There are no real roots |
The quadratic formula can
be derived by using the technique
of completing the square on
the general quadratic formula:
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Given: |
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Divide
through by a: |
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Move
the constant term to
the right side: |
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Add
the square of one-half
the coefficient of x
to both sides: |
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Factor
the left side (which
is now a perfect square),
and rearrange the right
side: |
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Get
the right side over
a common denominator: |
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Take
the square root of both
sides (remembering to
use plus-or-minus): |
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Solve
for x: |
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