F.4 A.maths Discriminant4

1. x2 + 2x + 3a = 0
D = 4 – 12a
D > 0 or 1/3 > a then real roots
a = 1/3 double roots
a > 1/3 complex roots
2. ax2 – 3x + 2a = 0
D = 9 – 8a2
D > 0 => 9 – 8a2 > 0
(3 – √8a)(3 + √8a) > 0
-3√2/4 < a < 3√2/4 then have real roots
If a = 3√2/4 or – 3√2/4 then double roots
-3√2/4 > a or a > 3√2/4 then complex roots
3. (a + 5)x2 – 3ax + (a – 5) = 0
D = 9a2 – 4(a + 5)(a – 5)
D = 9a2 – 4a2 + 100
D = 5a2 + 100 >0 must have real roots
4. x2 – 2ax + 2a2 + 1 = 0
D = 4a2 – 4(2a2 + 1)
D = - 4a2 – 4 < 0 must have complex roots
5. x2 – 4ax + 3a2 = 0
D = 16a2 – 4(3a2) = 4a2 > 0
Must have real roots. If a = 0, will have double roots