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# parabolas

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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.

Mar 19, 2021

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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 )

Mar 19, 2021