Question

How do you know when an equation has no real solution, just one solution, two solutions, or an infinite number of solutions? Give as many tips as possible.

Answer 1

use the b^2-4ac also called the discriminant to check
if this is negative --> no real solutions
positive --> two real distinct solutions
zero --> one real solution

Answer 2

think of the quad. formula:
\[x = \frac{ -b \pm \sqrt{b^2-4ac} }{ 2a }\]
if the discriminant is negaitve --> there is no real answer to a negative inside a radical.
positive --> well if you add a constant number to another two numbers (remember, plus-minus), there will be two solutions
zero --> if you add or subtract zero to any number, you get the same number.