Factorising quadratics

Quadratic equations are equations of the form $$ax^2 + bx + c = 0$$ where a, b and c are constants, $$a \ne 0$$ and $$x$$ is a variable. In other words, a quadratic equation has at least one term of the variable, say $$x$$, raised to the exponent $$2$$, e.g. $$x^2$$

Arranging terms
Arrange the quadratic into order: first the squared number ax2, then the number times x, bx, finally the constant value c.

Factorising quadratics
Form of quadratics: $$ax^2 + bx + c = 0$$

To factorise:
 * 1) split the middle term so it adds to the original number, e.g., let b = (AD + BC), and
 * 2) multiplies to the constant times the first term, e.g., Ax times Bx equals ABx2, then a = AB,
 * 3) then bracket so the pronumeral (letter) is like this, e.g., (Ax + C)(Bx + D).

Checking
Multiplying the two terms: $$(Ax + C)$$ and $$(Bx + D)$$ with each other becomes:

$$Ax \times Bx + Ax \times D + C \times Bx + C \times D$$

which rearranges to:

$$ABx^2 + (AD + BC)x + CD$$

The final constant $$c = CD.$$

Examples
$$2m^2 + 11m + 5$$

$$= (2m+1)(m+5)$$

To check it, re-expand the answer to see if we get back to where we started from:

$$(2M+1)(M+5)$$

$$= 2M \times M + 2M \times 5 + 1 \times M +1 \times 5$$

$$= 2m^2 + 11m + 5$$