Deriving The Quadratic Formula

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’.

An example of online maths tuition. The topic is deriving the quadratic formula using 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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