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ax2 + bx + c = 0
a,
b, c are constants (generally
integers)
Synonyms:
Solutions or Zeros
- Can
have 0, 1, or 2 real roots
Consider the graph of quadratic
equations. The quadratic equation
looks like ax2 + bx + c = 0,
but if we take the quadratic
expression on the left
and set it equal to y,
we will have a function:
y = ax2 + bx + c
When
we graph y vs. x,
we find that we get a curve
called a parabola.
The specific values of a,
b, and c control
where the curve is relative
to the origin (left, right,
up, or down), and how rapidly
it spreads out. Also, if a
is negative then the parabola
will be upside-down. What
does this have to do with
finding the solutions to our
original quadratic equation?
Well, whenever y = 0
then the equation y = ax2 + bx + c
is the same as our original
equation.
Graphically, y is
zero whenever the curve crosses
the x-axis. Thus, the
solutions to the original
quadratic equation (ax2 + bx + c
= 0) are the values of x
where the function (y = ax2 + bx + c)
crosses the x-axis.
From the figures below, you
can see that it can cross
the x-axis once, twice,
or not at all.
|  Actually,
if you have a graphing
calculator this technique
can be used to find
solutions to any
equation, not just
quadratics. All you
need to do is
- Move
all the terms to
one side, so that
it is equal to zero
- Set
the resulting expression
equal to y
(in place of zero)
- Enter
the function into
your calculator
and graph it
- Look
for places where
the graph crosses
the x-axis
Your graphing calculator
most likely has a function
that will automatically
find these intercepts
and give you the x-values
with great precision.
Of course, no matter
how many decimal places
you have it is still
just an approximation
of the exact solution.
In real life, though,
a close approximation
is often good enough. |
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