If the set of all solutions of |x^2 + x - 9| = |x| + |x^2 - 9| is [ , ] [ , ) , then ( ^2 + ^2 + ^2) is equal to:

Step-by-step solution

The given equation is |x^2 + x - 9| = |x| + |x^2 - 9| . Let A = x and B = x^2 - 9 . Then A + B = x^2 + x - 9 . The equation is of the form |A + B| = |A| + |B| , which holds true if and only if A B 0 . Substituting the values of A and B : x(x^2 - 9) 0 x(x - 3)(x + 3) 0 The critical points are -3, 0, 3 . Using the wavy curve method, the solution set is: x [-3, 0] [3, ) Comparing this with the given solution set [ , ] [ , ) , we get: = -3 , = 0 , = 3 We need to find the value of ^2 + ^2 + ^2 : ^2 + ^2 + ^2 = (-3)^2 +