SOLUTION: Does anyone have an idea of how quadratic equations can have the same solutions?
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Question 468457: Does anyone have an idea of how quadratic equations can have the same solutions?
Found 3 solutions by stanbon, solver91311, Theo:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Does anyone have an idea of how quadratic equations can have the same solutions?
----
y = 2(x-1)(x+2)
has the same solutions as
y = 5(x-1)(x+2)
----------
Cheers,
Stan H.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
I presume you mean "how two different quadratic equations can have the same solution"
Consider the quadratic equation
which has factors:
(x\ -\ \phi_2)\ =\ 0)
and therefore has the solutions 
and 
Now consider:
which has factors:
(x\ -\ \phi_2)\ =\ 0)
and solutions and
hence there are infinite quadratic equations with a pair of given solutions and where (x\ -\ \phi_2)\ =\ kax^2\ +\ kbx\ +\ kc\ =\ 0)
, one for every real number 
John
My calculator said it, I believe it, that settles it
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
They will have the same solution at the points where the graphs of each equation intersect each other.
An example would be:
x^2 + 5x + 3
-x^2 + 5x + 6
The graph of both of these equations is shown below:
The graph of these equations show that they will intersect at 2 places.
Those places would be the common solutions for both equations.
To find the intersection points, set the equations equal to each other and solve.
We start with:
x^2 + 5x + 3 = -x^2 + 5x + 6
Add x^2 to both sides of this equation and subtract 5x from both sides of this equation and subtract 6 from both sides of this equation to get:
2x^2 - 3 = 0
Add 3 to both sides of this equation to get:
2x^2 = 3
divide both sides of this equation by 2 to get:
x^2 = 3/2
Take the square root of both sides of this equation to get:
x = +/- sqrt(3/2)
The 2 equations should intersect at the same value of y when x = +/- sqrt(3/2).
That would be a common solution for both graphs.
Our 2 equations are:
y1 = x^2 + 5x + 3
y2 = -x^2 + 5x + 6
When x = + sqrt(3/2), these equations become:
y1 = 10.62372436
y2 = 10.62372436
x = + sqrt(3/2) is roughly located at x = 1.2
Look at the graph and you'll see that when x = 1.2, the 2 equations intersect at somewhere around y = 10.6.
When x = - sqrt(3/2), these equations become:
y1 = -1.623624357
y2 = -1.623724357
x = - sqrt(3/2) is roughly located at x = -1.2
Look at the graph and you'll see that when x = -1.2, the 2 equations intersect at somewhere around y = -1.6.
If you want to see what the graph of those equations looks like from a more distant perspective, it is shown below:
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