Let A, B , where A, B (- 2 , 2 ) , be the roots of the quadratic equation x^2 - 2x - 5 = 0 . Then 20 ^2 ( A+B 2 ) is equal to:

Step-by-step solution

Given that A and B are the roots of the quadratic equation x^2 - 2x - 5 = 0 . Sum of the roots: A + B = 2 Product of the roots: A B = -5 Using the tangent addition formula: (A+B) = A + B 1 - A B = 2 1 - (-5) = 2 6 = 1 3 Since A, B (- 2 , 2 ) and their product A B = -5 Since (A+B) = 1 3 > 0 , the angle A+B must lie in the first quadrant, i.e., A+B (0, 2 ) . Thus, (A+B) is positive. (A+B) = 1 1 + ^2(A+B) = 1 1 + ( 1 3 )^2 = 3 10 We need to find the value of 20 ^2 ( A+B 2 ) . Using the half-angle identity ^2 = 1 - (2