At what point(s) do lines y = x^2 - 5x - 14 and y = 3x + 2 intersect? Write your answer in surd form
To find the point(s) where these two lines intersect we will first find the x coordinate of the point(s) where they intersect snd use this to find the corresponding y coordinate by substituting the x value into one of the linear equations. To find the x value(s) we can use the fact that y = y, so we can write x2-5x-14 = 3x+2 since x2-5x-14 = y and 3x+2 = y. We can rearrange this to get x2-8x-16=0 which is a quadratic equation, meaning we can use the quadratic formula to find our x value(s).
You should find that there are two x values, x = 4 + 4(sqrt(2)) and x = 4 - 4(sqrt(2)) (sqrt(2) is square root 2!)We can now use these x values to find their corresponding y values simply by substituting them into y = 3x + 2.Doing this we find that the points where the two lines intersect are (4 + 4(sqrt(2)), 14 + 12(sqrt(2))) and (4 - 4(sqrt(2)), 14 - 12(sqrt(2))). To double check your values try substituting your x values into the other linear equation and see if they give you the same answer!
