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• Expressions of the type $$x^2 + bx + c$$ can be factorised by using the result $$(x + e) (x + f) = x^2 + (e + f) x + ef$$
Thus $$b$$ is the sum of two numbers $$(b = e+f$$) and $$c$$ is the product $$(c = e\times f$$)
• For example to factorise $$x^2 + 7x + 6$$, we need two numbers whose sum is equal to $$7$$ and product is equal to $$6$$
We select the appropriate numbers by guessing and checking. Solution is $$6$$ and $$1$$, giving the factors $$( x + 6)$$ and $$(x + 1)$$
• A similar process can be done when the coefficient of $$x^2$$ is a number other than $$1$$
• For example factorise $$6x^2 + 17x - 3$$: we need two numbers whose product is $$6$$, i.e. $$6$$ and $$1$$ or $$3$$ and $$2$$. We need two other numbers whose product is $$-3$$, i.e. $$3$$ and $$-1$$ or $$-3$$ and $$1$$. The correct combination of these give: $6x^2 + 17x - 3=(6x - 1) (x + 3)$