Question

The quadratic equation .x^2 + ax + b = 0 (a, b real numbers) has both roots real and positive if
and only if​
1) a<0 and b>0
2) ab<0 and a_> 2√b
3) ab<0 anda^2_>4b
4) b>0 and a_<-2√b

Answer 1

Given : The quadratic equation .x² + ax + b = 0 (a, b real numbers)  

To Find : both roots real and positive if and only if​  

1) a<0 and b>0

2) ab<0 and a_> 2√b

3) ab<0 anda^2_>4b

4) b>0 and a_<-2√b

Solution:

x² + ax + b = 0

root = m and n

real and positive

= > m  + n =   positive   and mn =   positive

m  + n =  - a    and mn  =  b

- a    is  positive if  a < 0

b is positive  if  b  > 0

also  a² - 4b  ≥ 0  for real roots

=>   a²  ≥ 4b

=>  a  ≤ - 2√b   as   a < 0

both roots real and positive if and only if​

 b > 0   and  a  ≤ - 2√b   this will automatically ensure a < 0

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