Factor pairs, then zero factors
For a monic quadratic, choose two numbers with the required sum and product. Collect zero form before using the null factor law
- Positive constant The factor constants have the same sign; their sum decides positive or negative
- Negative constant The factor constants have opposite signs; the larger magnitude determines the sign of their sum
- Solve the factors The constants in x+p and x+q have opposite signs to the corresponding roots
- Zero and repeated roots Do not divide away a factor x. List a repeated root once
- Check the original When the right side was nonzero, substitute back into that original equation, or state clearly that you are checking its collected zero form
Worked example
Factorise and solve the equation
- \(\displaystyle \begin{gathered}5+(-4)=1\\5(-4)=-20\end{gathered}\)Find a signed pair matching both coefficients
- \(\displaystyle (x+5)(x-4)=0\)Write the factorised zero form
- \(\displaystyle x=-5\text{ or }x=4\)Set each factor to zero
- \(\displaystyle (-5)^2+(-5)-20=0\)Check the smaller root
- \(\displaystyle 4^2+4-20=0\)Check the larger root
Common mistakes
Difficulty levels
Questions about this topic
What does monic mean?+
The coefficient of x² is 1. An equation with coefficient −1 can be made monic by multiplying every term by −1; its roots are unchanged
Why do the factor constants add and multiply?+
Expanding (x+p)(x+q) gives x²+px+qx+pq. The middle terms combine to (p+q)x.
How do I choose the signs?+
A positive product requires the same sign. A negative product requires opposite signs. Then use the required sum to decide which pair works
Why are factor signs opposite to roots?+
The factor x+p is zero when x=−p. Solve the short linear equation rather than copying the written sign
Does a repeated factor give two different roots?+
No. Both copies give the same value, so list one distinct root
What if there is no integer factor pair?+
Do not conclude there are no real roots. Another method may be needed, such as square roots, graphing or numerical approximation
Can different equations have the same roots?+
Yes. Multiplying an equation by a nonzero constant preserves its roots. An exact polynomial-factorisation task still requires the stated leading coefficient
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 10: Quadratic equations and graphs