Factorising Quadratics by Hand

• Factorise asks for an equivalent product of factors. For example, the answer to factorising $x^2+5x+6$ is $(x+2)(x+3)$. It does not ask for values of $x$.
• Solve by factorising asks you to write the product equal to zero and then find every root.
• Factorising reverses expansion. Expand your final brackets to check that every coefficient matches the original expression.

• If two numbers multiply to give zero, then at least one of them must be zero. There is no other way to reach zero by multiplying.
• In symbols: if $pq=0$, then $p=0$ or $q=0$.
• Factorising turns the quadratic into two brackets multiplied together. Then you can use the fact above on each bracket.

• These two roots are also the $x$-coordinates where the graph of $y=x^2+5x+6$ crosses the $x$-axis.

• A calculator can solve a quadratic equation, but it cannot replace the algebra when the question asks you to factorise an expression or show the factorisation.
• Factorising reverses expansion. This gives you a way to check every answer: expand the brackets and make sure you recover the original expression.

For $x^2+bx+c$, find two numbers $m$ and $n$ such that $mn=c$ and $m+n=b$. Then $x^2+bx+c=(x+m)(x+n)$.

This rule comes from expanding $(x+m)(x+n)=x^2+(m+n)x+mn$. The two numbers must create both the middle coefficient and the constant term, which is the number without $x$.

• For $x^2+5x+6$, use $2$ and $3$ because $2\times3=6$ and $2+3=5$. Therefore the factorisation is $(x+2)(x+3)$.
• For $x^2+2x-15$, use $5$ and $-3$ because $5(-3)=-15$ and $5+(-3)=2$. Therefore the factorisation is $(x+5)(x-3)$.

• If the product is positive, the two numbers have the same sign. If the product is negative, they have opposite signs.
• The required sum tells you which signed number must be farther from zero. For a positive sum, the positive number must be farther.