Find the coordinates where the curve f(X)= X^2+X-6 touches the x axis using factorisation.

The first part of this question requires be student to realise that touching the x axis equates to f(x) equalling 0, so we must equate the expression to 0.

Next we need to factorise the expression. This means we trying to find two numbers, a and b, which can be used to express the function in the form f(X)=(x-a)(x-b). If we multiply out this expression we see f(X)= X^2-(a+b)X+ab, so we need a+b=1 and ab=-6. Is we solve these simultaneously we find a=-3 b=2. If x=a or b we see one of the brackets equals 0 and since any number multiplied by 0 equals 0, So f(x) equals 0 (touches the x axis) when x= a or b.