The curve has equation y = x^3 - x^2 - 5x + 7 and the straight line has equation y = x + 7. One point of intersection, B, has coordinates (0, 7). Find the other two points of intersection, A and C.

As both equations are equal to y, we can combine them to create a single equation in terms of x: x^3 - x^2 -5X + 7 = x + 7. Shift the equation so the left hand side is equal to 0 on the right: x^3 - x^2 - 6x = 0. Now we can factorise out x to get a quadratic equation: (x)(x^2 - x -6) = 0. One of the values of x is 0, as we can see from the factor (x). Next we must factorise the remaining quadratic to get 3 seperate values of x. As the coefficient of the x^2 term is 1, we know both factors include an x. The negative constant (-6) tells us that the signs are different for both factors (i.e. there is one + and one -). Now we must apply different factors of the constant (6) to the two factors, to test which will give us the correct coefficient for the x term. This gives us (x)(x+2)(x-3) = 0 , which tells us that x = 0, x = 3 and x = -2. Then sub x = 3 and x = -2 (dont need x = 0 as this is point B) into y = x + 7, and we find y = 10, y = 5. Now simply match up the correct x and y values to get the coordinates of the points of intersection: A(-2,5), C(3,10).