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

Nov 12, 2016

There are 2 solutions.

#### Explanation:

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

Use $\Delta = \pm \sqrt{{b}^{2} - 4 a c} \text{ }$ from $\text{ } a {x}^{2} + b x + c = 0$

For $\text{ } - {x}^{2} + 5 x + 1 = 0$

$\Delta = \pm \sqrt{{\left(5\right)}^{2} - 4 \left(- 1\right) \left(1\right)}$
$= \pm \sqrt{25 + 4}$
$= \pm \sqrt{29}$

This indicates that there are 2 solutions.

(they exist, so they are Real but Irrational)

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

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