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?
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y = 2(x-1)(x+2)
has the same solutions as
y = 5(x-1)(x+2)
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Cheers,
Stan H.
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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