How to Find the Square Roots of 12-5i: A Step-by-Step Guide

Complex numbers are a fascinating topic in mathematics, and understanding how to find their square roots is crucial for advanced mathematical applications. In this article, we will explore a detailed method to find the square roots of the complex number 12 - 5i.

1. Introduction to Complex Numbers

A complex number is a number that can be expressed in the form z x yi, where x and y are real numbers, and i is the imaginary unit with the property i^2 -1. The complex number 12 - 5i is a specific example, where the real part is 12 and the imaginary part is -5.

2. The Method: Expressing in Standard Form

To find the square roots of 12 - 5i, we express it in the form z x yi. Let's assume that the square roots of 12 - 5i are z_1 x yi and z_2 a bi.

2.1 Squaring the Roots

We start with the equation:

(x yi)^2 12 - 5i

Expanding the left side, we get:

x^2 - y^2 2xyi 12 - 5i

By equating the real and imaginary parts, we obtain the following system of equations:

x^2 - y^2 12

2xy -5

2.2 Solving the System of Equations

From the second equation, we can express y in terms of x:

y -frac{5}{2x}

Substituting this into the first equation:

x^2 - (-frac{5}{2x})^2 12

which simplifies to:

x^2 - frac{25}{4x^2} 12

Multiplying through by 4x^2, we get:

4x^4 - 25 48x^2

Rearranging, we obtain a quartic equation:

4x^4 - 48x^2 - 25 0

2.3 Solving the Quadratic Equation

Let u x^2, then the equation becomes a quadratic:

4u^2 - 48u - 25 0

Using the quadratic formula, we find:

u frac{48 pm sqrt{(-48)^2 - 4 cdot 4 cdot (-25)}}{2 cdot 4} frac{48 pm sqrt{2304 400}}{8} frac{48 pm sqrt{2704}}{8} frac{48 pm 52}{8}

This gives two solutions:

u frac{100}{8} 12.5

u frac{-4}{8} -0.5

Since u must be non-negative, we discard u -0.5. Therefore:

x^2 12.5 Rightarrow x sqrt{12.5} frac{5sqrt{2}}{2}

2.4 Finding y

Using the relationship y -frac{5}{2x}, we get:

y -frac{5}{2 cdot frac{5sqrt{2}}{2}} -frac{1}{sqrt{2}} -frac{sqrt{2}}{2}

3. The Square Roots of 12 - 5i

We now have:

z_1 frac{5sqrt{2}}{2} - frac{sqrt{2}}{2}i

z_2 frac{5sqrt{2}}{2} frac{sqrt{2}}{2}i

4. An Alternative Method: Denesting Technique

Another approach involves denesting the radical. We assume:

sqrt{12 - 5i} sqrt{x} - isqrt{y}

Squaring both sides, we get:

12 - 5i x - y - 2isqrt{xy}

This gives us the first equation:

12 x - y

Multiplying the roots and setting them equal to 13:

sqrt{12 - 5i} cdot sqrt{12 5i} 13 sqrt{xy}

Therefore:

xy 169 Rightarrow x frac{25}{2}, y frac{1}{2}

Thus, the denested form is:

sqrt{12 - 5i} frac{5sqrt{2}}{2} - frac{isqrt{2}}{2}

The other root is the negative of this:

-sqrt{12 - 5i} -frac{5sqrt{2}}{2} frac{isqrt{2}}{2}

5. Conclusion

We have found the square roots of 12 - 5i in two different methods. Both approaches confirm that the square roots are:

boxed{frac{5sqrt{2}}{2} - frac{isqrt{2}}{2}} text{ and } boxed{-frac{5sqrt{2}}{2} frac{isqrt{2}}{2}}

This method can be generalized to find the square roots of any complex number, making it a valuable tool in complex number analysis. Understanding these methods not only enhances your problem-solving skills but also deepens your appreciation for the elegance of mathematics.