Question

Solve: x^(2)=-5+12i The solution in the first quadrant is: The solution in the third quadrant is:

Answer 1

The equation x^2 = -5 + 12i is solved by finding the square roots of the complex number -5 + 12i, leading to solutions in the first and third quadrants by considering the principal and negative square roots accordingly.

Step-by-step explanation:

The question asks to solve the equation x2 = -5 + 12i for the complex number x. The complex number solutions for a quadratic equation can be found using methods involving complex numbers, such as taking the square root of both sides. Since we are looking for solutions in specific quadrants, we need to find the principal square root (first quadrant) as well as the negative square root (third quadrant).

To solve the equation, we find the square root of the complex number on the right-hand side. The solution in the first quadrant, which has a positive real and imaginary part, can be determined using the square root of a complex number. Similarly, the solution in the third quadrant, which has negative real and imaginary parts, is simply the negative of the first quadrant's solution.

Using the complex square root method:

1. Express -5 + 12i in polar form.
2. Take the square root of the modulus and halve the argument to find the principal square root.
3. Apply the negative sign for the third quadrant solution.

This process will yield the solutions in the desired quadrants.