Find the square root of complex number 7+24iLearn how to find the square roots of complex numbers! This video covers two examples, showing you the step-by-step process of calculating square roots of complex numbers. Perfect for complex number beginners!
𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Find the square root of complex number 7+24i
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
Let the square root of 7 + 24i be x + yi, where x and y are real numbers.
Then, (x + yi)² = 7 + 24i
Expanding the left side, we get:
x² + 2xyi + (yi)² = 7 + 24i
x² + 2xyi - y² = 7 + 24i
(x² - y²) + (2xy)i = 7 + 24i
Equating the real and imaginary parts, we get two equations:
1) x² - y² = 7
2) 2xy = 24 = xy = 12
From the second equation, y = 12/x. Substitute this into the first equation:
x² - (12/x)² = 7
x² - 144/x² = 7
Multiply the entire equation by x²:
x⁴ - 144 = 7x²
x⁴ - 7x² - 144 = 0
This is a quadratic equation in x². Let u = x². Then the equation becomes:
u² - 7u - 144 = 0
Factor the quadratic:
(u - 16)(u + 9) = 0
So, u = 16 or u = -9.
Since x is a real number, x² must be non-negative. Therefore, x² = 16, which gives x = ±4.
Now find the corresponding values of y using y = 12/x:
If x = 4, y = 12/4 = 3.
If x = -4, y = 12/(-4) = -3.
Thus, the square roots of 7 + 24i are 4 + 3i and -4 - 3i. We can write this as ±(4 + 3i).