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)
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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.
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If a single number is "the solution" it must have multiplicity 2.
If x = 2 is the only solution,
y = a(x-2)^2
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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:

Apply FOIL

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:

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