Find the points at which the equation y = x^2 - 12x + 35 intersects the x-axis.

To find the points at which an equation intersects the x-axis, we first need to factorise the equation to be able to find the solutions. To do this, we consider the general form of a quadratic equation ax^2 + bx + c, and need to use the coefficient of x (b, which in our question = -12), along with the constant (c, which in our question = 35).

b and c are used to find two values p and q, such that p + q = b, and p * q = c.

Here, through inspection of the problem we can find the values p = -5 and q = -7. Double checking these:

b = p + q = (-5) + (-7) = -12 c = p * q = (-5) * (-7) = 35

Since we get matching values for b and c, then these are the correct values. We must now perform the factorisation step, and do so by writing the equation in the form y = (x + p)(x + q), giving us:

y = (x - 5)(x - 7)

We can now carry out the final step to find the points at which the curve intersects the x axis, and to do this we simply set y to 0 and solve for x. We have to find values of x such that (x - 5)(x - 7) = 0, and do this by setting each bracket = 0. This will leave us with two solutions as follows:

The first is when (x - 5) = 0, giving us a value of x = 5.

The second is found similarly by setting (x - 7) = 0, giving us a value of x = 7.

Therefore, the answer to the initial question is y = x^2 - 12x + 35 intersects the x axis when x = 5 and x = 7.

TIPS

(Note: These are designed to help you find a solution more quickly once you have an understanding of the principles involved)

When looking to factorise a quadratic equation, it may be helpful to check whether the sign before the c value is a plus or a minus, as this will indicate if the signs in the factorised expression are the same or different. If the sign before c is a -, then the factorised expression will have a different sign in each bracket and look similar to the form (x + p)(x - q). If it is a + however, then the signs will both be the same, and will also be the same sign as the one preceeding b (This can be seen in our example where we have c = +35 and b = -12, which gave a factorisation of the form (x - p)(x - q).)