The quadratic formula, derived

Adam Zettel · September 22, 2026 · written with an agent

A garden has a perimeter of 10 m and an area of 6 m². How long are its sides? The answer needs $x^{2} - 5 x + 6 = 0$ , and solving it by completing the square gives the quadratic formula.

Open in the workbench Download .vivOutputs below are from the site build’s run; Run all runs it again in your browser.

Look inside this program

Atlas
The program’s map: every function and which calls which.
There’s no recorded run of this program yet.

A rectangular garden has a perimeter of 10 m and an area of 6 m². How long are its sides?

Call one side $x$ . The two sides add up to half the perimeter, so the other side is $5 - x$ , and the area says

x (5 - x) = 6, that is, x^{2} - 5 x + 6 = 0.

You could guess $x = 2$ here, but for most areas a guess won't work, so we solve the general equation $a x^{2} + b x + c = 0$ instead, in three steps.

Complete the square

Factor $a$ out of the first two terms, then add and subtract $(b /2 a)^{2}$ inside:

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

That is $a (x - h)^{2} + k$ , with $h = - b /2 a$ and $k = c - b^{2} /4 a$ . The point $(h, k)$ is the vertex of the parabola.

For the garden, $h = 2.5$ and $k = - 0.25$ : a square garden, 2.5 m on a side, would have an area of $2. 5^{2} = 6.25$ m², which is 0.25 m² too much.

new: vertex

Rung 1 tests 1 of 1 pass

the vertex of $x^2 - 5x + 6$ is $(2.5, -0.25)$ (passes)

Solve from the vertex

Now $a (x - h)^{2} + k = 0$ is one step from solved. Move $k$ across and take the square root:

$(x - h)^{2} = - \frac{k}{a}, x = h \pm - \frac{k}{a} .$

For the garden, $0.25 = 0.5$ , so $x = 2.5 \pm 0.5$ and the sides are 2 m and 3 m.

Write $h$ and $k$ out in terms of $a$ , $b$ and $c$ : $- k / a = (b^{2} - 4 a c) /4 a^{2}$ , and the roots become the quadratic formula,

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

new: roots

Rung 2 tests 1 of 1 pass

the roots of $x^2 - 5x + 6$ are 2 and 3 (passes)

When there are no real roots

The formula takes the square root of the discriminant,

$D = b^{2} - 4 a c .$

If $D > 0$ there are two roots, if $D = 0$ one, and if $D < 0$ none: the parabola never reaches the axis. A garden with a perimeter of 10 m and an area of 7 m² gives $x^{2} - 5 x + 7 = 0$ and $D = 25 - 28 = - 3$ . No such garden exists. The square, at 6.25 m², is the largest area that perimeter can enclose.

new: discriminant

Rung 3 tests 1 of 1 pass

$x^2 - 5x + 7$ has no real roots (passes)

1Complete the square

2Solve from the vertex

3When there are no real roots

Complete the square

Solve from the vertex

When there are no real roots