Given the quadratic equation :

y= x^2 + 3x + b

We need to determine b such that y has two complex roots.

We know that if delta < 0, then the quadratic equation has 2 complex roots.

==> delta = b^2 - 4ac < 0

==> a = 1 b= 3 c = b

==> 9 - 4*1*b < 0

==> 9- 4b < 0

==> -4b < -9

==> b > 9/4

Then the values of b should be greater that 9/4

**Then, b belongs to the interval ( 9/4 , inf )**

A quadratic equation ax^2 + bx + c = 0 has two complex roots if b^2 - 4ac < 0

In the given equation x^2 + 3x + b,

a = 1 , b = 3 and c = b( to be determined)

Now as b^2 - 4ac = 9 - 4b < 0

=> 4b > 9

=> b > 9/4

**Therefore b > 9/4**

Q:What is b if y= x^2 + 3x + b has two complex roots.

Solution:

If a quadratic equation ax^2+bx+c = 0 has two complex roots, then the discriminant b^2-4ac < 0.

So if x^2 +3x+b = 0 has 2 roots, then 3^2 - 4*1 *b < 0.

=> 9 - 4b < 0.

=> 9 < 4b.

=> 9/4 < b.

=> b > 9/4.

**So**** x^2+3x+b = 0 will have two complex roots i****f b > 9/4.**