Question

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

Answer 1

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`.