SOLUTION: Hi, I need help with this question. Detailed working out would be greatly appreciated. Find the coordinates, in terms of k, of the points of intersection of the line with eq

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hi, I need help with this question. Detailed working out would be greatly appreciated. Find the coordinates, in terms of k, of the points of intersection of the line with eq      Log On


   

Question 1199676: Hi,
I need help with this question. Detailed working out would be greatly appreciated.
Find the coordinates, in terms of k, of the points of intersection of the line with equation y=x+k and the parabola with equation y=x^2-4x, where k>0. (3 marks)
This is what i have done so far:
x+k = x^2-4x
k = x^2-4x-x.
Thanks.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Your first equation is correct.
The second equation you wrote is valid, but it's not particularly useful.

We do not want to solve for k.
Instead, we want to isolate x.

The presence of the x^2 term tells us we'll need the quadratic formula.

First let's get everything to one side.
x+k = x^2-4x
0 = x^2-4x-x-k
x^2-4x-x-k = 0
x^2 + (-4x-x) - k = 0
x^2 + (-5x) - k = 0
1x^2 + (-5)x + (-k) = 0

The last equation mentioned is of the form ax^2+bx+c = 0
where,
a = 1
b = -5
c = -k

Let's plug those items into the quadratic formula.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-5%29%2B-sqrt%28%28-5%29%5E2-4%2A1%2A%28-k%29%29%29%2F%282%2A1%29

x+=+%285%2B-sqrt%2825%2B4k%29%29%2F%282%29

x+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29 or x+=+%285-sqrt%2825%2B4k%29%29%2F%282%29
Unfortunately we cannot simplify this any further.

Those are the possible x coordinates of the solutions of the form (x,y)

Let's plug the first x solution mentioned into y = x+k to find the corresponding y coordinate.
y+=+x%2Bk

y+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29%2Bk

y+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29%2B2k%2F2

y+=+%282k%2B5%2Bsqrt%2825%2B4k%29%29%2F%282%29

Similarly, plug the other x coordinate into that equation to find that y+=+%282k%2B5-sqrt%2825%2B4k%29%29%2F%282%29

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Answer:

The two points of intersection are:

and


Side note:
Since k > 0, we ensure that 25+4k > 0 meaning that the radicand is never negative.
We don't have to worry about applying the square root to a negative.