Question

if A and B are the root of equation X square + a x minus b​

Answer 1

Given equation x^2 +ax +b=0,

Sum of roots = a+b = -a,

 2a+b=0.

Product of roots= ab = b,

For non-zero format, a=1, b = -2.

Hence function is f(x) = x^2 +x -2.

so x =-1/2.

Putting this value in above f(x),

f(-1/2) = 1/4 -1/2 -2 = -9/4.

So minimum value will be -9/4.

Answer 2

if A and B are the root of equation X square + a x minus b​

A & B are the roots of X² + ax - b

As A & B are roots of polynomial

So equation is

(X - A)(X-B)

= X² - AX - BX + AB

=> X² -(A+B)X + AB

Equating with

X² + ax - b

A+B = -a

AB = - b

if We assume A & a are same thing & also B & b is same

then

ab = -b

=> a = -1

a + b = -a

=> -1 + b = 1

=> b = 2

Then X² + ax - b = X² -X -2

X² -X -2

= X² - 2X  + X - 2

= X(X-2) +1(X-2)

= (X+1)(X-2)

A = - 1 & B = 2