Solve by the null factor law
Write zero on one side, factorise the other side and solve every factor branch. This method directly solves factorisable quadratics; other quadratics may need a different method
- Zero is essential A nonzero product does not allow you to set its factors to zero \(\displaystyle (x+3)(x-2)=4\)
- Make zero form Perform the same subtraction on both sides \(\displaystyle \begin{aligned}x^2+x&=12\\x^2+x-12&=0\end{aligned}\)
- Keep all factor branches A factor x gives the root zero. Do not cancel it \(\displaystyle x(3x-2)=0\)
- Count distinct values A repeated factor gives one distinct root \(\displaystyle (x-5)^2=0\quad\Longrightarrow\quad x=5\)
- Check the original Substitute each root into the equation you started with, not just the factorised expression
Worked example
Solve the equation by factorisation
- \(\displaystyle 3x^2+3x-36=0\)Subtract 36 from both sides
- \(\displaystyle 3(x^2+x-12)=0\)Take out the common factor 3
- \(\displaystyle 3(x+4)(x-3)=0\)The factor constants multiply to −12 and add to 1
- \(\displaystyle (x+4)(x-3)=0\)Divide both sides by 3
- \(\displaystyle \begin{gathered}x+4=0\text{ or }x-3=0\\x=-4\text{ or }x=3\end{gathered}\)Solve each factor
- \(\displaystyle \begin{aligned}3(-4)^2+3(-4)&=36\\3(3)^2+3(3)&=36\end{aligned}\)Both solutions check in the original equation
Common mistakes
Difficulty levels
Questions about this topic
Does “or” mean both factors must be zero?+
No. Either factor being zero makes the product zero. Sometimes both factors are zero at the same input
Can I use the law when the product equals 4?+
No. Rearrange to zero form, then factorise that new expression
Why can I divide by 6 but not simply by x?+
Six is known to be nonzero. Dividing by x would exclude the possible value x = 0
What does a repeated root mean?+
The same factor occurs twice, but it gives only one distinct solution value
How are roots related to the graph?+
For y equal to the zero-form quadratic, roots are the x coordinates of its x-axis intersections. A repeated root touches the axis
What if the quadratic does not factor easily?+
Do not invent integer factors. Another method, such as completing the square, a quadratic formula or a numerical method, may be needed
How should I enter fractional roots?+
Use the requested ascending order. Where numerator and denominator are separate, use lowest terms and a positive denominator
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs