Final answer:
The quadratic equation x² + 81 = 0 has two complex solutions, 9i and -9i, which are the result of taking the square root of a negative number. No real number solutions exist for this equation.
Step-by-step explanation:
The equation provided x² + 81 = 0 is a quadratic equation of the form ax² + bx + c = 0. To solve for x, we need to find the values of x that satisfy the equation.
To do this, we can rearrange the equation and isolate the term x² by subtracting 81 from both sides:
x² = -81
Next, we take the square root of both sides of the equation. Since there is no real number that squared gives a negative result, the solutions will be complex numbers:
x = ±√(-81)
x = ± 9i
Therefore, the solutions for x are the complex numbers 9i and -9i. None of the provided options (a through d) are correct, as they all neglect the imaginary component.