Final answer:
The equation x^2 = 16 has two solutions, x = 4 and x = -4, because both the positive and negative roots are valid for a square root operation. However, x^2 = -16 has no real solutions because the square of a real number cannot be negative.
Step-by-step explanation:
Equation x^2 = 16 and x^2 = -16 do not have the same solutions because the basic rules of multiplication apply differently to positive and negative numbers. When two positive numbers or two negative numbers are multiplied together, the result is positive, as shown by the examples 2 x 3 = 6 and (-4) x (-3) = 12. However, equation x^2 = -16 suggests that a number squared would yield a negative result, which is impossible in the set of real numbers because squares of real numbers are always non-negative.
Equation x^2 = 16 has two solutions because a square root operation has both a positive and a negative root. This means that the two numbers which, when squared, equal 16, are 4 and -4. To solve the equation, we consider both the principal square root √16 = 4 and the negative square root -√16 = -4 as valid solutions. Therefore, the solutions to x^2 = 16 are x = 4 and x = -4.