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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.
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 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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