Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x-6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons.
+128475 Setting the y's equal we have
2x^2 - 10x - 10 = x^2 - 4x - 6 rearrange as
x^2 - 6x - 4 = 0 complete the square
x^2 - 6x + 9 = 4 + 9
(x - 3 )^2 = 13 take both roots
x - 3 = sqrt (13) x - 3 = - sqrt (13)
x = sqrt ( 13) + 3 x = 3 - sqrt (13)
Putting the first x value into x^2 -4x - 6 we get
(sqrt 13 + 3)^2 - 4(sqrt (13) + 3) - 6 =
(13 + 6sqrt 13 + 9) - 4sqrt 13 - 12 - 6 =
4 + 2 sqrt (13)
Putting the second x value into x^2 - 4x - 6 we get
(3 - sqrt (13)^2 - 4(3 - sqrt 13) - 6 =
9 - 6sqrt13 + 13 - 12 + 4sqrt 13 - 6 =
4 - 2sqrt 13
Solutions are
(3 - sqrt 13 , 4 - 2sqrt 13) ; ( 3 + sqrt 13, 4 + 2sqrt 13 )
