Here we have to find the sum and product of the roots of the equation 2x^2+5x+9=0. Instead of first calculating the roots and then adding and multiplying them, let us use Viete's relation:

We have the sum of the roots as -b/a and the product of the roots as c/a

Here a = 2, b= 5 and c= 9

Therefore the sum of the roots is -b/a = - 5/2 and the product of the roots is c/a = 9/2.

**The required sum and product of the roots are - 5/2 and 9/2 resp.**

2x^2+5x+9 = 0

To find the product of roots:

If x1 and and x2 are the roots of the equation , then by the relation between the roots and and the coefficients of the equation, we get:

x1+x2 = -5/2

x1*x2 = -9/2= -4.5.

Therefore the product of rootsÂ = -4.5.

2[x^2+2*5/4x+(5/4)^2]-2*(5/4)^2+9

=2(x+5/4)^2-2(25/16)+9

=2(x+5/4)^2+47/8

We have2(x+5/4)^2>=0 when x=-5/4

so 2(x+5/4)^2+47/8>=47/8

the minimum is 47/8 when x=5/4