Question

An equation x^2 -6x +18 = 0

The line y= 41 meets C at the point R.
Find the x-coordinates of R, giving you your answer in the form p + q(sqrt(2)), where p and q are integers.

I'm confused about this representation, I got -3 and 9 as the x coordinates yet I have no idea how to place it into this formula. The answer is 3 + 4(sqrt(2)) but I have no idea as to how to arrive at this. Do I use completed square form?

Answer 1

This representation is fine for your purposes - no need to use the completed square form.

To arrive at this solution, you should set the expression (x^2-6x+18) equal to 41, and solve the resulting quadratic. Subtract 41 from both sides to get x^2-6x-23, and from there, apply the quadratic formula to get:


`(6+-√(128))/(2)`, or
`3+-4√(2)`. The line intersects the parabola at two points,
`3+4√(2)` and
`3-4√(2)`.