# Solving Quadratic Equations by Factors

The factorised form of a quadratic equation is $(ax+b)(cx+d)=0$. Using the Null Factor law, we can solve this equation algebraically to find $x$

\begin{align} \displaystyle (ax+b)(cx+d) &= 0 \\ ax+b &= 0 \text{ or } cx+d =0 \\ x &= -\dfrac{b}{a} \text{ or } x = -\dfrac{d}{c} \end{align}

### Example 1

Solve $(x-1)(x+2)=0$.

\begin{align} \displaystyle (x-1)(x+2) &= 0 \\ x-1 &= 0 \text{ or } x+2 = 0 \\ \therefore x &= 1 \text{ or } x = -2 \end{align}

### Example 2

Solve $x(x-3)=0$.

\begin{align} \displaystyle x(x-3) &= 0 \\ x &= 0 \text{ or } x-3 = 0 \\ \therefore x &= 0 \text{ or } x = 3 \end{align}

### Example 3

Solve $(2x-1)(3x+2)=0$.

\begin{align} \displaystyle (2x-1)(3x+2) &= 0 \\ 2x-1 &= 0 \text{ or } 3x+2 = 0 \\ \therefore x &= \dfrac{1}{2} \text{ or } x = -\dfrac{2}{3} \\ \end{align}

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