Question

How many solutions does `-x^2+5x+1=0` have?

Answer 1

There are 2 solutions.

Explanation:

You can check the nature of the roots (solutions) without actually finding them.

Use `Delta = +-sqrt(b^2 -4ac)" "` from `" "ax^2 +bx+c =0`

For `" "-x^2 +5x+1 =0`

`Delta =+-sqrt( (5)^2 -4(-1)(1))`
`= +-sqrt(25+4)`
`=+-sqrt29`

This indicates that there are 2 solutions.

(they exist, so they are Real but Irrational)

(note: usually working with `-x^2 +5x+1 =0` is not very comfortable, but because it is an equation, it can be changed to
`x^2 -5x-1 =0` by multiplying by `-1`

(This does not change the outcome for the solutions.)