## 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!

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

Would you like help with any of the maths in this post?

## Trigonometric functions and the unit circle

The other day, a student asked me:

## “When I use the trig functions, what do the numbers on my calculator really mean?”

We discussed this question in detail and linked the output values from the SIN and COS calculator buttons for different angles to the horizontal and vertical measurements from the centre of the unit circle to a point P on the circumference. For example, when the angle $\theta$ is 0 degrees, the point P has coordinates (1, 0) and then when $\theta$ is 90 degrees, the coordinates are (0, 1).

By plotting the different values of the x and y coordinates for different points around the circumference of the circle, we were able to produce the graphs of the cosine and sine functions.

We also discussed the value of the tangent function as a ratio of the sine and cosine functions and why this leads to some of the more unsual features of the graph of tan ($\theta$).

Would you like help with any of the maths in this post?