I Wanted To Find The Square Root Of (1-2i)
But I Didn't Know How To Solve It.
My Solution :
x+yi = sqr(1-2i)
x2-y2+2xyi = 1- 2i
x2-y2 = 1
2xy = -2
y = - 1/x
x2 - (-1/x) = 1 ] * x2
x4-x2-1 = 0
How To Evaluated (x4-x2-1) ?
I Tried To Sol
I would convert the complex number to polar.
1-2i= sqrt5 @arctan-2=sqrt 5 @-63.4349488 degrees or @(360-63.43)deg figure that angle out.
sqrt (1-2i)= sqrt(sqrt5)@1/2 (that angle)
now convert back to polar.
lets do it so it can be checked.
(5^.25)=1.49534878
that angle= = 296.565051
1/2 that angle= = 148.282525
sqrt(1-2i)=1.4953cos 148.282525+i(1.4953sin 148.282525)
= -1.27-i*0.786
Now, just for fun sake,lets square that.
(-1.27^2)-(..786^2)-2i(1.21*.786)
= 1.61-.617 -i 1.90=.99-1.9i
Ok, you can work it with more accuracy to get it.
To solve the equation x^4 - x^2 - 1 = 0, you can use various methods such as factoring, the quadratic formula, or numerical methods like Newton's method or using a graphing calculator/computer program.
In this case, you can use a numerical method since the equation is a fourth-degree polynomial, which can be difficult to factor or solve algebraically.
One common numerical method to approximate the roots of a polynomial is Newton's method. Here's how you can use it to find the roots of x^4 - x^2 - 1 = 0:
1. Choose an initial guess, let's say x0 = 1. This is somewhat arbitrary, but it helps to start with a reasonable value close to where the root might be.
2. Use Newton's iteration formula to improve the initial guess:
- xn+1 = xn - f(xn) / f'(xn), where f(x) represents the given polynomial and f'(x) is its derivative.
3. Compute the value of the polynomial f(x) and its derivative f'(x) at each iteration:
- f(x) = x^4 - x^2 - 1
- f'(x) = 4x^3 - 2x
4. Repeat the iteration process until you reach an acceptable level of accuracy or until you find a root that satisfies the equation.
By performing these iterations, you will gradually converge to one or more approximate solutions for the equation x^4 - x^2 - 1 = 0.
Alternatively, you can also use numerical methods built into graphing calculators or computer programs like MATLAB or Python with appropriate libraries, which can directly find the roots of a polynomial equation without manual iteration.