Let = 2 ⁻¹ 1 5 + ⁻¹ 1 8 and = ⁻¹ 130 11 . If and are the roots of the quadratic equation ax^2 + bx + c = 0 , where a, b, c are integers, a > 0 , and (a,b,c)=1 , then the value of a + b + c is:

Step-by-step solution

First, we evaluate . = 2 ⁻¹ 1 5 + ⁻¹ 1 8 Using the identity 2 ⁻¹x = ⁻¹ ( 2x 1-x^2 ) : 2 ⁻¹ 1 5 = ⁻¹ ( 2/5 1 - 1/25 ) = ⁻¹ ( 2/5 24/25 ) = ⁻¹ 5 12 Now, add this to the second term: = ⁻¹ 5 12 + ⁻¹ 1 8 Using ⁻¹x + ⁻¹y = ⁻¹ ( x+y 1-xy ) : = ⁻¹ ( 5 12 + 1 8 1 - 5 12 1 8 ) = ⁻¹ ( 10+3 24 1 - 5 96 ) = ⁻¹ ( 13/24 91/96 ) = ⁻¹ ( 13 24 96 91 ) = ⁻¹ ( 4 7 ) Thus, = 4 7 . Next, we evaluate . = ⁻¹ 130 11 = 130 11 Using ^2 = ^2 - 1 : ^2 = 130 121 - 1 = 9 121 = 3 11 (since is in the principal range [0, /2) ). The roots of the qua