Find the complex number u=x+yi where x,y € I, such that u^2=-5+12i. HELPPPPPPP!
(x + yi)^2 = x^2 + 2xyi - y^2
x^2 - y^2 = -5
x y = 6
i would suggest a graphical solution
... try fooplot
u = 2 + 3i
To find the complex number u = x + yi, such that u^2 = -5 + 12i, we can use algebraic methods.
Let's substitute u = x + yi into the equation u^2 = -5 + 12i:
(x + yi)^2 = -5 + 12i
Expanding the left side of the equation using the binomial square formula:
x^2 + 2xyi + (yi)^2 = -5 + 12i
Now, simplify each term:
x^2 + 2xyi - y^2 = -5 + 12i
Since the equation must hold for any real x and y, we can equate the real and imaginary parts on both sides of the equation:
Equating the real parts:
x^2 - y^2 = -5 ... (1)
Equating the imaginary parts:
2xy = 12 ... (2)
Now, we have a system of equations consisting of equations (1) and (2). Let's solve them simultaneously:
From equation (2), we can isolate y:
y = (12 / 2x)
y = 6 / x
Substitute this value of y into equation (1):
x^2 - (6 / x)^2 = -5
Simplifying:
x^4 - 36 = -5x^2
Rearranging terms:
x^4 + 5x^2 - 36 = 0
This is now a quadratic equation in terms of x^2. We can solve it by factoring or using the quadratic formula.
Let's factor the equation:
(x^2 - 4)(x^2 + 9) = 0
Setting each factor equal to zero:
x^2 - 4 = 0 or x^2 + 9 = 0
Solving for x:
x^2 = 4 or x^2 = -9
Taking the square root of each:
x = ±2 or x = ±3i
Now, we have the possible values for x. Let's substitute them back into y = 6 / x to find the corresponding y values:
For x = 2:
y = 6 / 2 = 3
For x = -2:
y = 6 / (-2) = -3
For x = 3i:
y = 6 / (3i) = 2i
For x = -3i:
y = 6 / (-3i) = -2i
Therefore, we have four possible complex numbers u:
u1 = 2 + 3i
u2 = -2 - 3i
u3 = 3i + 2i = 5i
u4 = -3i - 2i = -5i
So, the complex numbers u, where u = x + yi and u^2 = -5 + 12i, are u1 = 2 + 3i, u2 = -2 - 3i, u3 = 5i, and u4 = -5i.