Equations » Derivation of the quadratic formula

Below is explained how the quadratic formula / abc-formula is derived. Looking for the explanation and / or examples of the quadratic formula? You can find them at the theory about quadratic equations.

How do you work?

Every quadratic equation can be written as ax² + bx + c = 0. As you want to know what x is, you should bring everything to the right-hand side of the =-sign. To get this, you shall have to take the square root of both sides to get rid of the square.
To get only x² on the left is impossible. For that reason you try to write the left-hand side as p² + 2pq + q². With the rule p² + 2pq + q² = (p + q)² you will get a nice square of which you can take the square root.
So you have to write the left hand side as p² + 2pq + q² so you are able to write it as (p + q)². Take the square root of both sides and islolate the x and simplify the right hand side as far as possible.

The derivation

`1:`	`ax`² + `bx` + `c`	= 0
`2:`	`ax`² + `bx`	= –`c`
`3:`	`x`² + `bax`	= –`ca`
`4:`	`x`² + 2`b`2`ax`	= –`ca`
`5:`	`x`² + 2`b`2`ax` + `b`2`a`	= `b`2`a` – `ca`
`6:`	`x` + `b`2`a`	= `b`2`a` – `ca`
`7:`	`x` + `b`2`a`	= `b`²4`a`² – 4`ac`4`a`²
`8:`	`x` + `b`2`a`	= `b`² – 4`ac`4`a`²

`9:`	`x` + `b`2`a` =	or `x` + `b`2`a` = –
`10:`	`x` = –`b`2`a` + 2`a`	or `x` = –`b`2`a` – 2`a`
`11:`	`x` = –`b` + 2`a`	or `x` = –`b` – 2`a`

Bij substituting b² – 4ac for D you will get (D = Discriminant):

x = –b + square root(D) 2a or x = –b – 2a (with D = b² – 4ac)

Explanation of the steps

`1:`	Initial equation.
`2:`	Do – `c` on both sides.
`3:`	Divide everything by `a`.
`4:`	Rewrite `ba` to 2`b`2`a` so you second term will be 2`pq`.
`5:`	Do + `b`2`a` on both sides so you get `q`² as the third term.
`6:`	Use `p`² + 2`pq` + `q`² = (`p` + `q`)².
`7:`	Simplify `b`2`a` and change `ca` so the denominator is 4`a`².
`8:`	Make one fraction.
`9:`	Take the square root on both sides. Two solutions!
`10:`	Do – `b`2`a` on both sides and use on the right-hand side.
`11:`	Final answer.