Quadratic equation: the discriminant in 5 seconds

The universal method

To solve $ax^2 + bx + c = 0$ (with $a \ne 0$), compute the discriminant:

$\Delta = b^2 - 4ac$

Then look at the sign of $\Delta$:

Case	Number of solutions	Formula
$\Delta > 0$	Two distinct solutions	$x_{1,2} = \dfrac{-b \pm \sqrt{\Delta}}{2a}$
$\Delta = 0$	One double solution	$x_0 = \dfrac{-b}{2a}$
$\Delta < 0$	No real solution	—

Example: $x^2 - 5x + 6 = 0$

• $a = 1$, $b = -5$, $c = 6$.
• $\Delta = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1$.
• $\Delta > 0$ → two solutions.
• $x_1 = \dfrac{5 + 1}{2} = 3$, $x_2 = \dfrac{5 - 1}{2} = 2$.

The key move

Always write down $a$, $b$, $c$ before computing $\Delta$. This is where 90% of sign mistakes happen.

Verification trick

Sum of roots = $-b/a$, product of roots = $c/a$. A quick check on your work.

On the example: $x_1 + x_2 = 5$ and $-b/a = 5$ ✓. $x_1 \times x_2 = 6$ and $c/a = 6$ ✓.

?Your turn

How many solutions does 2x² − 4x + 5 = 0 have?

A.Two distinctB.One doubleC.No real solutionD.Infinitely many

Check