1. Quadratic Theory Introduction This algebraic expression: x 2 + 2x + 1 is a polynomial of degree 2 Expressions like this in which the highest power of x is x 2 are also called quadratics. Quadratic Equations look like this: What values of x make the quadratic expression 0? What are the roots of this quadratic equation?

2. Solving Quadratic Equations Quadratic equations may be solved by:  Factorising  Completing the Square  Using the quadratic formula

3. Solve x 2 – 2x – 8 = 0 using each of the above methods  Factorising When factorising quadratics check for:  common factors  difference of 2 squares  double brackets

4.  Completing the Square

5.  Using the quadratic formula Reminder: the roots are

6. The Discriminant x 2 – 6x + 10 = 0x 2 – 6x + 9 = 0x 2 – 6x + 8 = 0 Solve Sketch Roots x = 2 or x = 4 x = 3 x = ? Equation

7. As we can see from the above examples, the nature of the roots of a quadratic can be found using the discriminant  Ifthere are two distinct roots  If there are two identical roots (i.e. one root)  If there are no real roots Examples:

8. Tangents to Curves To determine whether a straight line cuts, touches or does not meet a curve the equation of the line is substituted into the equation of the curve. When a quadratic equation results, the discriminant can be used to find the number of points of intersection.

9. Two Points of Intersection 2 distinct real roots 2 distinct points of intersection

10. One Point of Intersection 1 real root 1 point of intersection Line is a tangent to the curve

11. No Points of Intersection no real roots no points of intersection The line and the curve do not intersect

12. Examples involving Tangents to Curves:

13. Quadratic Inequalities A quadratic inequality is an expression such as: The problem is to find the values of x for which such an expression is true A quadratic inequality can be solved using a sketch of the quadratic function We can then easily see where the graph is positive or negative.

14. Find the values of x for which: First sketch the curve. y intercept at: (0,12) x intercepts at: (1½,0) and (-4,0) a) From the graph 12 – 5x – 2x 2 is positive (i.e. below the x axis) when -4 < x < 1½ b) From the graph 12 – 5x – 2x 2 is negative when x < -4 or 1½ < x (i.e. above the x axis)

15. Examples involving solution of Inequalities: