The Discriminant and Nature of Roots

The discriminant is the expression under the square root in the quadratic formula:

$$\Delta = b^2 - 4ac$$

It tells you how many real solutions the equation $ax^2 + bx + c = 0$ has, without solving:

Why it works: in $x = \frac{-b \pm \sqrt{\Delta}}{2a}$:

- If $\Delta > 0$: $\sqrt{\Delta}$ is a real positive number $\to$ two different values from $\pm$
- If $\Delta = 0$: $\sqrt{0} = 0$ $\to$ $\pm 0$ gives only one value
- If $\Delta < 0$: $\sqrt{\text{negative}}$ is not real $\to$ no real solutions

Bonus: if $\Delta$ is a perfect square ($1, 4, 9, 16, ...$), the solutions are rational (nice fractions). If not, they contain square roots.