Quadratic equations of the form

$$x^2+bx+c=0$$

can be factorised to create an equation of the form

$$(x+m)(x+n)=0\text{.}$$

When we expand a factorised quadratic expression we obtain

$$(x+m)(x+n)=x^2+(m+n)x+(m \times n)\text{.}$$

To factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.

a)

$$\simplify{x^2+{v1+v2}x+{v1*v2}=0}$$

We need to find two values that add together to make $\var{v1+v2}$ and multiply together to make $\var{v1*v2}$.

$$\begin{align} \var{v1} \times \var{v2}&=\var{v1*v2}\\ \var{v1}+\var{v2}&=\var{v1+v2}\\ \end{align}$$

So the factorised form of the equation is

$$\simplify{(x+{v1})(x+{v2})}=0\text{.}$$

b)

We can begin factorising by finding factors of $\var{v3*v4}$ that add together to give $\var{v3+v4}$.

$$\begin{align} \var{v3} \times \var{v4}&=\var{v3*v4}\\ \var{v3}+\var{v4}&=\var{v3+v4}\\ \end{align}$$

So the factorised form of the equation is

$$\simplify{(x+{v3})(x+{v4})}=0\text{.}$$

c)

When factorising the quadratic expression

$$\simplify{x^2+{v5*v6}=0}$$

we need to find two values that add together to make $0$ and multiply together to make $\var{v5*v6}$.

$$\begin{align} \var{v5} \times \var{v6}& = \var{v5*v6}\\ \simplify[]{ {v5} + {v6}} &= 0 \\ \end{align}$$

So the factorised form of the equation is

$$\simplify{(x+{v5})(x+{v6})}=0\text{.}$$