Solving Quadratic Equations by Factors


The factorise form of a quadratic equation is $(ax+b)(cx+d)=0$. We can solve this equation algebraically to find $x$ by using the Null Factor law.

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

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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