F.4 A.maths Discriminant7

1. x^2+5x+3+a=0
△ = 5^2 - 4(3+a) = 13 - 4a
when 13 - 4a > 0, i.e. a < 13/4 ,it have two real roots.
when 13 - 4a = 0 , i.e. a = 13/4, it have double roots = - 5 /2
when 13 - 4a < 0 , i.e. a > 13/4, it have no real root.
2)x^2+(a+b)x+ab=0
△ = (a+b)^2 - 4ab = a^2 + b^2 +2ab - 4ab
= (a - b)^2 >= 0
when (a-b)^2 > 0 , it have two real roots.
when (a-b)^2 = 0 , i.e. a = b = 0 , it have double roots = 0.
3. (b-c)x^2+(c-a)x+(a-b)=0 has a repeated real solution, so
△ = (c-a)^2 - 4(b-c)(a-b) = 0
c^2 + a^2 - 2ac = 4(ab - ac - b^2 + bc)
c^2 + a^2 -2ac = 4ab + 4bc - 4b^2 - 4ac
c^2 + a^2 + 2ac = 4(a+c)b - 4b^2
4b^2 - 4(a+c)b + (a+c)^2 = 0
(2b)^2 - 2(a+c)(2b) + (a+c)^2 = 0
[2b - (a+c)]^2 = 0
2b - (a + c) = 0
b = (a + c) / 2