Question

How many and of what type are the solutions to x2 + 9x + 10 = 0?

No real solutions

Two identical rational solutions

Two different rational solutions

Two irrational solutions

Answer 1

We would check the discriminant.
D = b^2 - 4ac

The value of the discriminant will determine our solutions

To have no real solutions, the discriminant will be negative
Two identical rational solutions occur when the discriminant is 0
Two different rational solutions occur when we have a positive discriminant that is a perfect square (examples: 4^2 = 16, 7^2 = 49, etc.)
Two irrational solutions occur when we have a positive discriminant that is not a perfect square.

Answer 2

you can see how these properties work out when looking at the quadratic formula
\[ \frac{-b \pm \sqrt{\color{goldenrod}{b^2 - 4ac}}}{2a}~~<- \color{goldenrod}{discriminant}~in~color\]
The discriminant is under that square root
if the discriminant is negative, then the square root is not real
if the discriminant is 0, then the square root becomes 0 and we end up adding and subtracting 0, which is the same number both ways
if the discriminant is postive and perfect square, then the square root will disappear and we'll just have a rational number
and, if the discriminant is positive and not perfect, then that square root will stick around, which makes an irrational number.