2. Reminders (1) The graph of a quadratic function is a parabola. If a > 0 the parabola isshaped and the turning point is a minimum. If a < 0 the parabola isshaped and the turning point is a maximum.

3. (2) To sketch and annotate a parabola we need to identify where possible : whether the shape is(a > 0)or(a < 0) the co-ordinates of the y-axis intercept (0, c) the roots (zeros) of the function by solving the equation of the axis of symmetry the co-ordinates of the turning point c Note to find the axis of symmetry when there are two roots x 1 and x 2 carry out the following :-

4. (3) When the equationis written in the formthe axis of symmetry is x = - p and the turning point is (-p, q) x = -p (-p, q)

5. (4) Quadratic equations may be solved by using the graph factorising completing the square using the quadratic formula (see point (5) below) (5)

6. The Discriminant (Nature of the roots of a quadratic Equation) Example Solve the following equations using the quadratic formula (-2) 2 - 4 x 4 x (-1) = 4 - (-16) = 20 Roots are given by : Two distinct real roots Graphical Implications x x Or

7. 4 2 - 4 x 4 x 1 = 16 - 16 = 0 Roots are given by : (twice) One real root (repeated/equal roots) Graphical Implications x x OR

8. (-2) 2 - 4 x 4 x 1 = 4 - 16 = -12 Roots are given by : Since  is not a real number there are NO REAL ROOTS Graphical Implications x x OR

9. Conclusion :- (i) positive2 real roots (ii) zero1 real root (iii) negative no real roots discriminates between real and non-real roots and so is called the DISCRIMINANT SUMMARY (i) the roots are real and distinct (unequal) TWO ROOTS (ii) the roots are real and equal ONE ROOT (iii) the roots are non-real NO ROOTS

10. Example 1 Determine the nature of the roots of the following equations:- (-6) 2 - 4 x 1 x 8 = 36 - 32 = 4 Sinceroots are real and distinct (-6) 2 - 4 x 1 x 9 = 36 - 36 = 0 Sinceroots are real and equal

11. (-6) 2 - 4 x 1 x 10 = 36 - 40 = -4 Sinceroots are non-real Example 2 Note This could be distinct roots or equal roots. Solution For real roots 4 2 - 4 x 2 x p  16 - 8p  16  p 8p  16 p  2 So equation has equal roots when p  2

12. Example 3 Find the value of q so thathas equal roots. Solution Cross-multiplying gives For equal roots So So equation has equal roots when

13. Example 4 Given that k is a real number, show that the roots of the equation are always real numbers. Solution Sinceis a square it has minimum value 0, thereforeand so the roots of the equation are always real.

14. Example 5 For what values of p does the equation x 2 - 2px + (2 - p) =0 have non-real roots? Solution x 2 - 2px + (2 - p) =0 For non-real roots (-2p) 2 - 4 x 1 x (2 - p) < 0 4p 2 - 8 + 4p < 0 4p 2 + 4p - 8 < 0 4(p 2 + p - 2) < 0 This is a quadratic INEQUATION. To solve this carry out the following steps :-

15. Step 1 Factorise the quadratic So 4(p 2 + p - 2) < 0 can be written as4(p + 2)(p - 1) < 0 Step 2 Next consider solving 4(p + 2)(p - 1) = 0 So p = - 2 or p = 1 Step 3 Then consider other values of p either side of these two values of p and also between these values :- setting working out in a table as follows:-

16. Table of Values To solve 4(p + 2)(p - 1) < 0 p (p + 2) (p - 1) -21 4(p + 2)(p - 1) -3 02 0 - 0 + 0 0 - - + + - - + + + So 4(p + 2)(p - 1) < 0 for values of p in the range -2 < p < 1 So the equation has non-real roots when -2 < p < 1