## Lecture

\[ 6x^2 - x - 12 \]

\[ 6x^2 -9x +8x -12 \]

Factorise the first two and last two terms separately:

\[ 3x(2x -3) +4(2x -3) \]

Notice the common factor of \( 2x-3 \) and factorise again:

\[ (2x-3)(3x+4) \]

## Seminar 1

\[x^2 + 8x \equiv (x+4)^2 -16 \]

Suppose we want to express the quadratic expression \(x^2 + 8x + 10 \) in completed square form? Well, we can write:

\[ \begin{align}

x^2 + 8x +10 \equiv (x+4)^2 -16 + 10 \\

\equiv (x+4)^2 -10

\end{align} \]

The reason the method is called "completing the square" comes from the geometric representation of this process. Click here for more detail.

## Seminar 2

*no real solutions*is what students are usually taught in their prior education).

## Seminar 3

- Make a table of x and y values for the function
- Plot the points on graph paper
- Draw a smooth curve through the points.