1. TheDiscriminant

2. What is to be learned?  What the discriminant is  How we use the discriminant to find out how many solutions there are (if any)

3. The Big Nasty Formula x = -b + √b 2 – 4ac 2a - How many solutions? 1.x 2 + 6x + 9 = 0 2.x 2 + 8x + 9 = 0 3.x 2 + 4x + 9 = 0 120120

4. The Big Nasty Formula x = -b + √ b 2 – 4ac 2a - How many solutions? 1.x 2 + 6x + 9 = 0 2.x 2 + 8x + 9 = 0 3.x 2 + 4x + 9 = 0 120120 The Discriminant tells you how many solutions you have zero positive negative

5. The Discriminant b 2 – 4ac If b 2 – 4ac is positive If b 2 – 4ac is zero If b 2 – 4ac is negative  2 solutions  1 solution  0 solutions

6. The Discriminant b 2 – 4ac If b 2 – 4ac is positive If b 2 – 4ac is zero If b 2 – 4ac is negative  2 solutions  1 solution  0 solutions Or

7. The Discriminant b 2 – 4ac If b 2 – 4ac > 0 If b 2 – 4ac = 0 If b 2 – 4ac < 0  2 solutions  1 solution  0 solutions Or  2 real roots  1 real root  no real roots (equal roots)

8. Nature of Roots x 2 + 5x – 11 = 0 c.f. ax 2 + bx + c = 0 a = 1, b = 5, c = -11 Using Discriminant b 2 – 4ac =5 2 – 4(1)(-11) =25 + 44 = 69 2 real roots

9. The Discriminant The b 2 – 4ac part of the BNF If b 2 – 4ac > 0 If b 2 – 4ac = 0 If b 2 – 4ac < 0  2 solutions  1 solution  0 solutions ( 2 real roots) (1 real root) (no real roots) (equal roots)

10. Nature of Roots x 2 + 6x + 10 = 0 c.f. ax 2 + bx + c = 0 a = 1, b = 6, c = 10 Using Discriminant b 2 – 4ac =6 2 – 4(1)(10) =- 4 no real roots

11. Equal Roots? 2x 2 - 8x + g = 0 c.f. ax 2 + bx + c = 0 a = 2, b = -8, For equal roots (-8) 2 – 4(2)g = 0 (-8) 2 – 4(2)g = 0 64 – 8g = 0 64 – 8g = 0 64 = 8g g = 8 g = 8 b 2 – 4ac = 0 -8g = -64 g = -64 -8 g = 8 c = g

12. Equal roots? 2x 2 + (m+1)x + 8 = 0 c.f. ax 2 + bx + c = 0 a = 2, b = m+1, c = 8 For equal roots (m+1) 2 – 4(2)(8) = 0 (m+1) 2 – 4(2)(8) = 0 m 2 + 2m + 1 m 2 + 2m + 1 m 2 + 2m – 63 = 0 (m + 9)(m – 7) = 0 m = -9 or m = 7 m = -9 or m = 7 b 2 – 4ac = 0 (m+1)(m+1) =m 2 +2m+1 - 64 = 0

13. No real roots? 2x 2 - 8x + g = 0 c.f. ax 2 + bx + c = 0 a = 2, b = -8, c = g For no real roots (-8) 2 – 4(2)g < 0 64 – 8g < 0 64 – 8g < 0Inequation b 2 – 4ac < 0

14. Solving Inequations quick reminder 4x - 2 10 4x 12 x 3 x 3 + 2 = > > > One big difference (Solution)

15. 10>6 Add 4 10 + 4> 6 + 4 10 + 4> 6 + 4 Multiply by 3 10X3 > 6X3 10X3 > 6X3 Divide by 2 10÷2 > 6÷2 10÷2 > 6÷2 Divide by -2 10÷(-2) 6÷(-2) 10÷(-2) 6÷(-2) -5-3 > >< If dividing/multiplying by a negative you must turn sign round

16. Back to…. 64 – 8g < 0 – 8g < -64 – 8g < -64 g 8 g 8 No real roots if g > 8 >