Solve for x: Understanding Absolute Values in |x²+4| = 40

To solve the equation $|x^2 + 4| = 40$, we consider two cases based on the properties of absolute value:

• Case 1: $x^2 + 4 = 40$
  - Subtract 4 from both sides to isolate the quadratic term:
  $x^2 = 40 - 4$
  $x^2 = 36$
  - Solving for $x$, take the square root of both sides:
  $x = \pm \sqrt{36}$
  $x = \pm 6$

• Case 2: $x^2 + 4 = -40$
  - Subtract 4 from both sides:
  $x^2 = -40 - 4$
  $x^2 = -44$
  - Since $x^2 = -44$ has no real number solutions (as the square of a real number cannot be negative), we discard this case.

Therefore, the only valid solutions are from Case 1: $x = \pm 6$.

Thus, the solution to the equation $|x^2 + 4| = 40$ is $x = \pm 6$.