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

2. Example 2
Return to main:

3. Example 3
Return to main:

4. 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)

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

6. 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:

7. 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)

8. 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