SOLUTION: Find the range of k in the of equations below if this system has 2 real solutions. Show your work. y=(x-1)^2+3 & y=2x+k

Question 1071488: Find the range of k in the of equations below if this system has 2 real solutions. Show your work. y=(x-1)^2+3 & y=2x+k
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
--> --> --> -->
If we chose a real so that has 2 real solutions for ,
we will have the corresponding real solutions for .
For a quadratic equation to have 2 real solutions
(which could both be the same)
it must be .
For the two real solutions to be different we need .
So, in the case of ,
where , and ,
to have 2 real solutions (which could both be the same) we must have
--> --> --> --> .
As an interval it would be

.

If we wanted to have two different real solutions,
we would need to have ,
and in interval notation that would be

.

IF YOU WERE STUDYING CALCULUS,
the answer would be obvious, if somewhat hard to explain the reasoning.
You would know that the slope of the tangent to
is ,
and since the slope of is ,
the slope of both functions would be the same when
<--> <--> .
That point could be the point of tangency if
both functions have the same value for .
That would happen for a real such that at <-->
, meaning that

<---> .
That point of tangency would be (2,4), with .
For <--> ,
the function has at .
For both functions would have the at

If function will have , and will be passing through a point inside the parabola, crossing it twice:
If function will have , and will be below the point (2,4) where the slope of is ,
and since the slope of increase with increasing x,
the function , with its constant will never be able to catch up.

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