Factorize Quadratic Equation

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Theory:
A quadratic expression is one where the highest power on the variable is 2. Here are some examples of quadratic expressions:

1.  x²,
2. z² + 3z,
3.  2x² + 3x - 5,
4. a² - 1,
5.  p² - 4p - 6

When these quadratic expressions are factorised, they are written as the product of two factors.
This is how to factorise these quadratic expressions:
Example 1: x²

Write as a product: x×x

Example 2: z² + 3z

Take 'z' out as a common factor: z(z + 3)

Example 3: 2x² + 3x - 5

This expression can be factorised if the middle term: '3x', is replaced by '-2x+5x', like this:
  2x² - 2x + 5x - 5
See the end of this article for an explanation of how to split the middle term and write down the factorised quadratic immediately without further steps.
After doing this, the first two terms and the last two terms can be factorised:
  2x(x - 1) + 5(x - 1)
Now there are only two terms, and there is a common factor: (x-1), so the expression can be factorised again, like this:
  (x - 1)(2x + 5)

Example 4: a² - 1

This expression can be expressed as: a² - 1² which is a difference of perfect squares, a special pattern that you need to be able to recognise. It can be written immediately as: 
  (a + 1)(a - 1)
In general, any expression of the form:  a² - b²
can then be written immediately as:  (a + b)(a - b).

Example 5: p² - 4p - 6

In this case, the middle term cannot be split to allow factorising as in example 3.
You can use Algematics to change this expression to a difference of perfect squares, as in example 4 above:
  
then factorise like this:
  
which simplifies to: