Consider the equation (2+ 3 )^ x + (2- 3 )^ x = 52 . If the sum of all solutions of this equation in the interval x [0, 4 ] is k , then the value of k is equal to

Step-by-step solution

Let u = (2+ 3 )^ x . Since (2+ 3 )(2- 3 ) = 1 , the equation becomes: u + 1 u = 52 u^2 - 52u + 1 = 0 u = 52 2704 - 4 2 = 26 15 3 We can express 26 15 3 as a perfect cube. Consider (2 3 )^3 : (2 3 )^3 = 2^3 3(2^2)( 3 ) + 3(2)( 3 )^2 ( 3 )^3 = 8 12 3 + 18 3 3 = 26 15 3 Thus, u = (2 3 )^3 . This implies (2+ 3 )^ x = (2+ 3 )^3 or (2+ 3 )^ x = (2- 3 )^3 = (2+ 3 )⁻³ . Equating the exponents, we get: x = 3 or x = -3 x = 1 3 or x = - 1 3 Let = ⁻¹ ( 1 3 ) , where 0 In the interval [0, 4 ] , the solutions for x = 1 3 are: x