Using The Discriminant Example 1 Find the nature of the roots of Two identical roots (=3) Return to main:

Example 2 Return to main:

Example 3 Return to main:

Tangents to Curves Example 4 Prove that the line y = 2x – 1 is a tangent to the parabola y = x2 and find the point of intersection. Parabola and line intersect where: 1 point of intersection Line is a tangent to the parabola Intersection is point (1,1)

Example 5 Find the equation of the tangent to y = x2 + 1 that has gradient 2. Line with gradient 2 is: Line and curve intersect where: 1 point of intersection Equation of tangent is:

Example 6 Find the equations of the tangents from (0, -2) to the curve y = 8x2 Line through (0, -2) has form: Line meets parabola where: Line is tangent to parabola: Equations of tangents are:

Solving Inequalities Example 7 Solve: First sketch the curve. y intercept at: (0,0) x intercepts at: (0 ,0) and (5,0) From the graph (i.e. on or below the x axis)

Example 8 Solve: First sketch the curve. y intercept at: (0,0) x intercepts at: (0 ,0) and (1,0) Where does y = -1 intercept the curve? line touches curve (tangent) at ( ½, -1) Solution: for all values of x