AHH complex numbers how to find roots for this equation?

In summary, to find the four roots of the equation z^4 + 7 -24i = 0, you can use De Moivre's theorem to write it in exponent form and then compare the coefficients of the real and imaginary parts to solve for the roots. This is similar to solving a polynomial equation by setting it equal to zero.
  • #1
meee
87
0
find the four roots of the equation
z^4 + 7 -24i = 0

completely lost, some help please...
 
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  • #2
ohhhh

z^4 = -7 + 24i...
= 25cis *** something
 
  • #3
MANY WAYS TO DO THIS.you can write it as a exponent.
z^4 + 7 -24i = 0
z^4 = - 7 +24i

Z = Cos(2*n*pi+ theta) + i Sin(2*n*pi + theta) = exponent(i*(theta+2*n*pi))
n = 0, 1,2,3,
so
Z^4 = exponent(i*(theta+2*n*pi)*4)
= Cos((theta+2*n*pi)*4) + i Sin((theta+2*n*pi)*4) (De Moivre's theorem)now you can compare the coefficeints of real /imaginary parts to find theta.
YOU SHOULD GET A SET OF SOLUTIONS. 4 unique as you substitiute n=0,1,2,3. then they'll repeat for n>3).
 
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  • #4
I bet you know how to solve x^4 - 17 = 0. Why not do the same thing for your problem?
 
  • #5
Ohhh yeah... thanksss! i think i mighta got it jolly... not sure coz i used cis not exponent

hurkyl... reali? because the 'i' made it confusing
 
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  • #6
hurkyl... reali? because the 'i' made it confusing
Yes, really! (7 - 24i) is just a number. Give it a name, like c, if it helps. and z^4 + (7-24i) = 0 (i.e. z^4 + c = 0) is just an ordinary polynomial equation.

The only difference (at least for this problem) is that it takes more work to simplify an expression involving a root.
 

1. How do I find the roots of a complex number equation?

The roots of a complex number equation can be found by setting the equation equal to 0 and then using the quadratic formula. However, since complex numbers have both real and imaginary parts, there will be two solutions for the roots.

2. What is the AHH complex number?

The AHH complex number, also known as the imaginary unit, is represented by the letter "i" and is defined as the square root of -1. It is used to represent the imaginary part of a complex number.

3. Can I find complex number roots without using the quadratic formula?

Yes, there are other methods for finding complex number roots such as using the polar form or the De Moivre's formula. However, the quadratic formula is the most commonly used method.

4. How do I know if the roots of a complex number equation are real or imaginary?

If the discriminant of the equation is negative, then the roots will be imaginary. If the discriminant is positive, then the roots will be real. If the discriminant is 0, then the roots will be equal and real.

5. Can all complex number equations be solved for roots?

No, not all complex number equations have solutions. For example, if the equation only has a real part or if the discriminant is negative, then the equation has no solution. However, most complex number equations can be solved using the quadratic formula.

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