C. 0.25 because you divide 0.06 by 0.24
Answer:
This is a function and it's because there's only one input for every output.
Step-by-step explanation:
Note: Although 5 and 1 both point to the same number this doesn't take away the validity of the function. :)
Complete the square.
![z^4 + z^2 - i\sqrt 3 = \left(z^2 + \dfrac12\right)^2 - \dfrac14 - i\sqrt3 = 0](https://tex.z-dn.net/?f=z%5E4%20%2B%20z%5E2%20-%20i%5Csqrt%203%20%3D%20%5Cleft%28z%5E2%20%2B%20%5Cdfrac12%5Cright%29%5E2%20-%20%5Cdfrac14%20-%20i%5Csqrt3%20%3D%200)
![\left(z^2 + \dfrac12\right)^2 = \dfrac{1 + 4\sqrt3\,i}4](https://tex.z-dn.net/?f=%5Cleft%28z%5E2%20%2B%20%5Cdfrac12%5Cright%29%5E2%20%3D%20%5Cdfrac%7B1%20%2B%204%5Csqrt3%5C%2Ci%7D4)
Use de Moivre's theorem to compute the square roots of the right side.
![w = \dfrac{1 + 4\sqrt3\,i}4 = \dfrac74 \exp\left(i \tan^{-1}(4\sqrt3)\right)](https://tex.z-dn.net/?f=w%20%3D%20%5Cdfrac%7B1%20%2B%204%5Csqrt3%5C%2Ci%7D4%20%3D%20%5Cdfrac74%20%5Cexp%5Cleft%28i%20%5Ctan%5E%7B-1%7D%284%5Csqrt3%29%5Cright%29)
![\implies w^{1/2} = \pm \dfrac{\sqrt7}2 \exp\left(\dfrac i2 \tan^{-1}(4\sqrt3)\right) = \pm \dfrac{2+\sqrt3\,i}2](https://tex.z-dn.net/?f=%5Cimplies%20w%5E%7B1%2F2%7D%20%3D%20%5Cpm%20%5Cdfrac%7B%5Csqrt7%7D2%20%5Cexp%5Cleft%28%5Cdfrac%20i2%20%5Ctan%5E%7B-1%7D%284%5Csqrt3%29%5Cright%29%20%3D%20%5Cpm%20%5Cdfrac%7B2%2B%5Csqrt3%5C%2Ci%7D2)
Now, taking square roots on both sides, we have
![z^2 + \dfrac12 = \pm w^{1/2}](https://tex.z-dn.net/?f=z%5E2%20%2B%20%5Cdfrac12%20%3D%20%5Cpm%20w%5E%7B1%2F2%7D)
![z^2 = \dfrac{1+\sqrt3\,i}2 \text{ or } z^2 = -\dfrac{3+\sqrt3\,i}2](https://tex.z-dn.net/?f=z%5E2%20%3D%20%5Cdfrac%7B1%2B%5Csqrt3%5C%2Ci%7D2%20%5Ctext%7B%20or%20%7D%20z%5E2%20%3D%20-%5Cdfrac%7B3%2B%5Csqrt3%5C%2Ci%7D2)
Use de Moivre's theorem again to take square roots on both sides.
![w_1 = \dfrac{1+\sqrt3\,i}2 = \exp\left(i\dfrac\pi3\right)](https://tex.z-dn.net/?f=w_1%20%3D%20%5Cdfrac%7B1%2B%5Csqrt3%5C%2Ci%7D2%20%3D%20%5Cexp%5Cleft%28i%5Cdfrac%5Cpi3%5Cright%29)
![\implies z = {w_1}^{1/2} = \pm \exp\left(i\dfrac\pi6\right) = \boxed{\pm \dfrac{\sqrt3 + i}2}](https://tex.z-dn.net/?f=%5Cimplies%20z%20%3D%20%7Bw_1%7D%5E%7B1%2F2%7D%20%3D%20%5Cpm%20%5Cexp%5Cleft%28i%5Cdfrac%5Cpi6%5Cright%29%20%3D%20%5Cboxed%7B%5Cpm%20%5Cdfrac%7B%5Csqrt3%20%2B%20i%7D2%7D)
![w_2 = -\dfrac{3+\sqrt3\,i}2 = \sqrt3 \, \exp\left(-i \dfrac{5\pi}6\right)](https://tex.z-dn.net/?f=w_2%20%3D%20-%5Cdfrac%7B3%2B%5Csqrt3%5C%2Ci%7D2%20%3D%20%5Csqrt3%20%5C%2C%20%5Cexp%5Cleft%28-i%20%5Cdfrac%7B5%5Cpi%7D6%5Cright%29)
![\implies z = {w_2}^{1/2} = \boxed{\pm \sqrt[4]{3} \, \exp\left(-i\dfrac{5\pi}{12}\right)}](https://tex.z-dn.net/?f=%5Cimplies%20z%20%3D%20%7Bw_2%7D%5E%7B1%2F2%7D%20%3D%20%5Cboxed%7B%5Cpm%20%5Csqrt%5B4%5D%7B3%7D%20%5C%2C%20%5Cexp%5Cleft%28-i%5Cdfrac%7B5%5Cpi%7D%7B12%7D%5Cright%29%7D)
Answer:
2
Step-by-step explanation:
Given the system of equations:
![x^2+y^2=9\\9x+2y=16](https://tex.z-dn.net/?f=x%5E2%2By%5E2%3D9%5C%5C9x%2B2y%3D16)
Comparing
with the general standard equation of a circle
.
The first equation is an <u>equation of a circle centred</u> at (0,0) with a Radius of 3.
The second equation 9x+2y=16 is a <u>straight line equation.</u>
A straight line can only intersect a circle at a maximum of 2 points.
Therefore the greatest possible number of solutions to the equations in the system is 2.