Deriving The Quadratic Formula | Matt Witchalls Maths Tuition

Ever wondered where the quadratic formula comes from?

Starting with the general form of a quadratic equation, you can derive the quadratic formula yourself using the technique of ‘completing the square’.

As you can see in the above screenshot from a recent online maths lesson, it is possible to start with the general form of a quadratic equation $ax^2 + bx + c = 0$ and manipulate it into the classic quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

The key to this derivation is the technique of completing the square. This is applied to the content of the red box in Line 2 and allows us to form the expression in Line 3.

Line 4 is all about making $(x + \frac{b}{2a})^2$ the subject and then creating an algebraic common denominator of $4a^2$ for the expression on the right-hand side.

Line 5 involves taking the square root of both sides and then $x$ is made the subject in the green equation in the bottom right corner.

Next, the box in the top right corner shows the square root being applied to the denominator of $4a^2$ to create a new common denominator of $2a$ . Finally, the two algebraic fractions are combined and we’re done!

$ax^2 + bx + c = 0 \rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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