The Quadratic Formula and the Discriminant Explained

Q: What does the discriminant tell you?

Its sign gives the number of real solutions: positive means two, zero means one (repeated), and negative means none.

Q: Does the quadratic formula always work?

Yes, for any quadratic equation with a ≠ 0. When the discriminant is negative it tells you there are no real solutions.

How to solve quadratic equations by factorising and with the quadratic formula, where the formula comes from, and how the discriminant predicts two, one or no real roots, seen on the graph.

3 min read Supports 3 lessons

A quadratic equation $ax^2 + bx + c = 0$ asks where the parabola $y = ax^2 + bx + c$ meets the $x$ -axis. Those crossing points are the equation's roots or solutions.

Solving by factorising

Zero product property

AB = 0 \;\Longrightarrow\; A = 0 \;\text{ or }\; B = 0

If a product is zero, at least one of its factors must be.

Worked example

Solve $x^2 + x = 12$ .

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Rearrange so one side is zero.
$x^2 + x - 12 = 0$

Watch out

Never divide both sides by $x$ . In $x^2 = 5x$ that throws away the solution $x = 0$ . Rearrange and factorise instead: $x(x - 5) = 0$ .

The formula that always works

Many quadratics do not factorise neatly. Completing the square on the general equation $ax^2 + bx + c = 0$ produces a formula that solves every one of them.

Quadratic formula

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

$2x^2 + 3x - 4 = 0$
Read off the coefficients.

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Where the formula comes from

$ax^2 + bx + c = 0$

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The discriminant

Everything under the square root, $b^2 - 4ac$ , is called the discriminant, often written $\Delta$ . You cannot take the real square root of a negative number, so its sign decides how many real solutions there are.

Discriminant

\Delta = b^2 - 4ac

$\Delta > 0$ : two real roots · $\Delta = 0$ : one repeated root · $\Delta < 0$ : no real roots

Same shape, different heights: the discriminant counts the crossings.

Tip

Check the discriminant first. If it is a perfect square, the quadratic factorises neatly and that is usually quicker.

Watch out

The whole of $-b$ is divided by $2a$ , not just the square root. Write one long fraction bar.

Common questions

What does the discriminant tell you?: Its sign gives the number of real solutions: positive means two, zero means one (repeated), and negative means none.
Does the quadratic formula always work?: Yes, for any quadratic equation with $a \neq 0$ . When the discriminant is negative it tells you there are no real solutions.

Practise quadratic formula

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