Maths explained

The quadratic formula, explained without memorising

Where the formula comes from, why the discriminant matters, and how to never forget it again.

Samuel Eludiora
8 Sept 2026, 1 min read

Most students memorise the quadratic formula. Understanding where it comes from makes it much harder to forget.

Start from completing the square

Take $a x^{2} + b x + c = 0$ and divide by $a$ :

x^{2} + \frac{b}{a} x = - \frac{c}{a}

Add $(\frac{b}{2 a})^{2}$ to both sides:

(x + \frac{b}{2 a})^{2} = \frac{b ^{2} - 4 a c}{4 a ^{2}}

Take square roots and rearrange:

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

What the discriminant tells you

The expression under the root, $b^{2} - 4 a c$ , decides everything:

Positive: the graph crosses the x-axis twice.
Zero: it just touches the axis.
Negative: it never reaches the axis, so there are no real roots.

Practise

Use the discriminant to decide how many roots $3 x^{2} - 2 x + 5 = 0$ has before solving anything. (It has none, because $4 - 60 < 0$ .)

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