Quadratic Equations

What is a quadratic?

A quadratic equation has the form:

$ax^2 + bx + c = 0 \quad (a \neq 0)$

Its graph is a parabola — a U-shape (opens up if $a > 0$, down if $a < 0$).

Method 1: Factoring

If you can write $ax^2 + bx + c$ as a product $(x - r_1)(x - r_2)$, then the solutions are $x = r_1$ and $x = r_2$ (zero product property).

Example. Solve $x^2 - 5x + 6 = 0$.

Find two numbers that multiply to $6$ and add to $-5$: those are $-2$ and $-3$.

$(x - 2)(x - 3) = 0 \implies x = 2 \text{ or } x = 3$

Method 2: The quadratic formula

When factoring doesn't work cleanly, use the quadratic formula. It always works:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example. Solve $2x^2 + 3x - 2 = 0$. Here $a = 2$, $b = 3$, $c = -2$.

$x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}$

So $x = \frac{2}{4} = \frac{1}{2}$ or $x = \frac{-8}{4} = -2$.

The discriminant

The piece under the square root, $b^2 - 4ac$, is the discriminant. It tells you how many real solutions the equation has before you finish solving:

Discriminant	Real solutions
$b^2 - 4ac > 0$	2 distinct real solutions
$b^2 - 4ac = 0$	1 repeated real solution
$b^2 - 4ac < 0$	0 real solutions (2 complex)

Key takeaways

• Try factoring first — it's fastest when it works.
• The quadratic formula never fails for $ax^2 + bx + c = 0$.
• The discriminant predicts how many real solutions exist.