Question

For what value of `p` does the quadratic `x^2-2px+p+2=0` have exactly one solution?

Answer 1

`p=-1` or `p=2`

Explanation:

The roots of a quadratic `ax^2+bx+c=0` are equal when the discriminant (`b^2-4ac`) is equal to `0`

In this case
`color(white)("XXX")a=1`
`color(white)("XXX")b=-2p`
`color(white)("XXX")c=(p+2)`

So the discriminant is
`color(white)("XXX")(-2p)^2-4(1)(p+2)`
`color(white)("XXX")=4p^2-4p-8`
which factors as
`color(white)("XXX")=4(p-2)(p+1)`

and the discriminant `=0` if
`color(white)("XXX")p=2` or `p=-1`