Y=x^(2)-2x-5 y=x^(3)-2x^(2)-5x-9 can you write the steps to find the intersection?

1 Answer
Teddy
Jun 14, 2018

The two equations intersect at (4,3).

Explanation:

First, we want to find the x-coordinate of the point(s) of intersection. We do this by setting the two equations equal to each other.
#x^3-2x^2-5x-9=x^2-2x-5#
#x^3-3x^2-3x-4=0#

To solve this equation, we can use the rational root theorem and try substituting #x=+-1,+-2,+-4# using synthetic division. We see that #x=4# is a solution and the remainder from division is #x^2+x+1#.
#(x-4)(x^2+x+1)=0#

Since the discriminant of #x^2+x+1# is -3, we know that polynomial has no real solutions. Therefore, #x=4# is the x-coordinate of the single point of intersection. To find the y-coordinate, just substitute #x=4# into any of the two equations to get that #y=3#.

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