Formula

Quadratic formula

Also written: quadratic equation formula · solving quadratics formula

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

The quadratic formula gives both solutions of any equation in the form ax² + bx + c = 0, whether or not it factors.

What each part means

$a$: the coefficient of x², and never zero
$b$: the coefficient of x
$c$: the constant term
$\pm$: read the formula twice, once with plus and once with minus

When to use it

The equation is quadratic and either does not factor or would take too long to try. It always works, so it is the fallback when factoring stalls.

Before you use it

Rearrange the equation so one side is zero. The formula reads $a$ , $b$ and $c$ off the standard arrangement $ax^2 + bx + c = 0$ , and reading them off anything else gives the wrong answer.

Keep the signs with the numbers. In $2x^2 - 5x + 3 = 0$ , the value of $b$ is $-5$ .

Where the formula comes from

It is completing the square, done once on the general equation instead of every time on numbers.

Start from $ax^2 + bx + c = 0$ , divide by $a$ , then move the constant across:

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

Add the square of half the $x$ coefficient to both sides:

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

The left side is now a perfect square, and the right side goes over a common denominator:

\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}

Take the square root of both sides, which is where $\pm$ appears, and subtract $\tfrac{b}{2a}$ :

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

Full walkthrough in completing the square.

Worked examples

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

Here $a = 1$ , $b = -5$ , $c = 6$ .

x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}

So $x = 3$ or $x = 2$ .

Solve $2x^2 + 3x - 1 = 0$ .

Here $a = 2$ , $b = 3$ , $c = -1$ .

x = \frac{-3 \pm \sqrt{9 + 8}}{4} = \frac{-3 \pm \sqrt{17}}{4}

This one does not factor over the whole numbers, which is exactly when the formula earns its place.

The part under the root

$b^2 - 4ac$ is the discriminant. Computing it first tells you how many real solutions to expect, and warns you before you take the root of a negative.

Lessons that teach this

Square Roots: What They Mean and How to Estimate Them
What a square root is, why √25 means 5 and not ±5, how to recognise perfect squares, and how to estimate a root like √50 to one decimal place without a calculator.
The Quadratic Formula: How and Why It Works
Learn the quadratic formula, where it comes from, and how to use it to solve any quadratic equation, with worked examples and practice problems.
Completing the Square, Step by Step
Learn completing the square: why the method works geometrically, how to handle a leading coefficient, and how it produces vertex form and the quadratic formula.