1. How to determine the number of solutions.
Let's look at the quadratic formula:
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":
x-2 = 0 or x-3 =0
Solving these we get:
x=2 or x=3
Now what you want is to be able to do this in reverse. Well all the steps above are

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