The key is the expression in the square root:
. In general there
are three cases:
1.
is positive. The square root of a positive number is also some
positive number. So in the numerator of the quadratic formula we will get
two values: (b + the square root) and (b  the square root). So
when
we get two solutions.
2.
is zero. The square root of zero is zero. So in the numerator
we get (b + 0) and (b  0). But both of these are equal to b! So
when
we only get one solution.
3.
is negative. And what is the square root of a negative
number? What can we square and get a negative number as an answer?
Answer: Nothing. You cannot square any Real number and get a negative.
So when
there are no solutions.
2. Finding the equation from the solution(s)
One way to find solutions from the equation is to factor it. For example, solving
we factor it:
For a product to be zero one of the factors must be zero. In other "words":
x2 = 0 or x3 =0
Solving these we get:
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x=2 or x=3
Now what you want is to be able to do this in reverse. Well all the steps above are
reversible. Therefore, if we have two solutions:
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 Spring '08
 PETRUSKA
 Quadratic Formula

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