Which of the following is not a root of z^2= 1-square root of 3i to the nearest hundredth.
a. -0.22+1.24i
b. -0.97-0.81i
c. 1.02-0.65i
d. 1.18-0.43i
Iam not sure how to do this problem. Can someone help me? Thanks
The magnitude of the right side is 4 so the magnitude of z must be 2. That is not true of any of the choices, so none of the choices are roots (possible values of z).
You can reach the same conclusion by squaring each choice and seeing what you get for the real and imaginary terms. None come close to 1-i*sqrt3. For example,
(-0.22+1.24i)^2 = (-0.22)^2 -(1.24)^2
-2*(0.22)(1.24)i = -1.489 -0.546 i
another way of doing this is to convert to polar
1-isqrt3=4@arctan-3/1
sqrt above=+- [email protected]
arctan -1.5=-56deg appx
so roots are
2@-56deg and 2@124deg or
2cos56-i2sin56 or 2cos124+isin124 or
1.18-1.65i or -1.18 +i1.65
which indicates none are roots.
To find the roots of the equation z^2 = 1 - √3i, we can solve for z by taking the square root on both sides.
To find the square roots, we can utilize the formula known as the square root of a complex number.
The formula states that if z = a + bi, where a and b are real numbers, then the square root of z is given by:
√z = ±(√((a + √(a^2 + b^2))/2) + i(√((√(a^2 + b^2) - a)/2))
Now, let's substitute the given equation z^2 = 1 - √3i into the formula:
z = ±(√((1 + √(1 + 4√3))/2) + i(√((√(1 + 4√3) - 1)/2))
Next, compute the values for z by performing the necessary calculations:
z1 = √((1 + √(1 + 4√3))/2) + i(√((√(1 + 4√3) - 1)/2))
z2 = -√((1 + √(1 + 4√3))/2) - i(√((√(1 + 4√3) - 1)/2))
Now, compare the calculated values of z with the provided options: -0.22 + 1.24i, -0.97 - 0.81i, 1.02 - 0.65i, and 1.18 - 0.43i.
If any of the calculated values match exactly with one of the given options, that option would be the root of the equation. If none of the calculated values match, you can use the distance formula to find the closest option to the calculated values to the nearest hundredth.
By comparing the calculated values to the given options, we find that none of the options match exactly. So, we need to find the closest option to the calculated values.
To do this, calculate the distance between each option and the calculated values using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Compare the distances and select the option with the smallest value as the nearest match.
After comparing the distances, you can determine the option that is not a root of z^2 = 1 - √3i by choosing the option with the smallest distance to the calculated values.