A curve has the equation y = 4x^2 + 5x + 3 and a line has the equation y = x + 2. Show that the line and the curve have one point of intersection.

Set the equations equal to each other: 4x^2 + 5x + 3 = x + 2Collect terms and set equal to 0: 4x^2 + 4x + 1 = 0Factorise the equation: (2x + 1)(2x + 1) = 0Can now find the value of x: 2x + 1 = 0, therefore 2x = -1, therefore x = -1/2As only one root is found, there must only be one point of intersection between the curve and the line. They intersect at the point x = -1/2
Alternatively, x = y - 2 can be substituted in. This will find the singular y coordinate of y = 3/2