If ( ⁻¹ x )^2+ ( ⁻¹ x )^2= 5 ^2 8 , then x^2+1=

Step-by-step solution

Let A = ⁻¹ x . Using the identity ⁻¹ x + ⁻¹ x = 2 , we have ⁻¹ x = 2 - A . Substituting into the equation ( ⁻¹ x)^2 + ( ⁻¹ x)^2 = 5 ^2 8 yields: A^2 + ( 2 - A )^2 = 5 ^2 8 Expanding gives: A^2 + ^2 4 - A + A^2 = 5 ^2 8 Simplifying and combining terms: 2A^2 - A + 2 ^2 - 5 ^2 8 = 0 2A^2 - A - 3 ^2 8 = 0 Multiplying through by 8: 16A^2 - 8 A - 3 ^2 = 0 Solving this quadratic using the formula: A = 8 64 ^2 + 192 ^2 32 A = 8 16 32 Yielding solutions A = 3 4 and A = - 4 . Since the principal range of ⁻¹ x is (- 2 , 2 ) ,