SOLUTION: How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution?
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Question 444589: How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution?
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
How do you find a quadratic equation if you are only given the solution?
If 2 and 3 are solutions the equation is
y = a(x-2)(x-3)
--------
Is it possible to have different quadratic equations with the same solution?
But "a" may be any non-zero number so there are lots of
Quadratic equations with the same solutions.
-----------
If a single number is "the solution" it must have multiplicity 2.
If x = 2 is the only solution,
y = a(x-2)^2
==================
Cheers,
Stan H.
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Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Let's assume that 
and 
are the roots of some quadratic equation 
Then with a little algebra we can see that 
and 
.
Reversing the Zero Product Rule, we can write:
\left(\sigma{x}\ -\ \beta\right)\ =\ 0)
Apply FOIL
x\ +\ \alpha\beta\ =\ 0)
Then let 
, 
, and 
and substitute:
The only fly in the ointment is that you can take any real number, say 
and apply it as a factor:
\left(\sigma{x}\ -\ \beta\right)\ =\ 0)
You still have the same roots to the equation, but when you multiply it out you get:
Which means that there are an infinite number of quadratic equations with a given pair of roots (or given root with a muliplicity of 2) but they only differ by a factor common to all three terms in the trinomial representation of the quadratic.
John
My calculator said it, I believe it, that settles it
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