If |x = 3i| <5, then prove that the complex number z also satisfies the inequality i(z+1)+1| <25.

Q: 4. Consider the complex equation - -32i. (a) Solve for all values of z.

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Q: Find all complex solutions of x - 3x+4 = 0. (If there is more than one solution, separate them with…

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Q: solve the complex equation using demoviers theorem x' – 1 = 0

A: Given equation is  x^7 -1=0

Q: 2 find all complex solutions of x = =-4+4/3i. 口 0..

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Q: Let , ² be the complex cube roots of unity. Let z₁, Z₂, Z3 be complex numbers such that Z₁ + Z₂ +Z₁…

A: Given that, Let ω,ω2 be the complex cube roots of unity. Let z1,z2,z3 be complex numbers such that…

Q: 5. Determine the roots of the complex numbers studied in class. z³ = 2 – 3i

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Q: Find all (real or complex) numbers z1 and z2 such that Z1+zz=2+4i Z1z2=-10–20i

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Q: Find all complex solutions of 3.x-6x+1 =0. (If there is more than one solution, separate them with…

A: Given: 3x2-6x+1=0 To Find: All the complex solution of the given equation. Concept: i. If…

Q: Prove that there is no complex number z such that 1 |z|«anda,z' = 1, where|a,| < 2. 3 r=1

A: In this question we have to prove that there is no complex number z such that z&lt;13 and ∑r=1n…

Q: Determine the four fourth roots of the complex number Z 3-4j, using De Moivre's theorem and express…

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Q: Determine all complex roots of h(x)=x3−2x+4 given that x=−2 is a root of the function.

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Q: Find the roots of complex numbers √√64 e¹.4i if n=4, k= 0,1,2,3 k=0, z1= k=1,z2= k=2, z3= k=3, z4=

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Q: (b) Use Gaussian Elimination to solve: X1 - 2 x2 - X3 + 3x4 = 4 2x1 + X2 + X3- 4x4 3 3x1 - X2 2x3 +…

A: Apply gaussian Elimination to the augmented matrix and back substituting.

Q: Find all complex square roots of 1+V3 i. In other words, find all complex solutions of x=1+/3 i.

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Q: Find the argument of the complex number 3/3 – 3i in the interval 0 <0< 2. Express your answer in…

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Q: Exercise 6.4.1 Show that 1 +i,2+i are the only two roots to P(x) =x² – (3+2i).x+(1+3i) Hence complex…

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Q: Prove that if the complex number z is a root of polynomial f(x) with real co-ffiecents , then z bar…

A: Solution

Q: Show that if z1 and z2 are complex numbers, then zi + z2 = Zī + Z2.

A: Suppose the two complex numbers are z1 and z2 such that where a,c are real parts and b,d are…

Q: 1- The distance between two complex numbers z1 Z2 is denoted by: a) Iz – z2| b) /x + y c) Vz, + z2…

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Q: According to the Fundamental Theorem of Algebra, the graph of fx) = -x2 - 2x +1, ha v two complex…

A: According to question We have to calculate the number of complex roots using fundamental theorem of…

Q: For complex number z and w, prove that zw-wz=z - w if and only if z =w or, zw = 1.

A: We know that , for a complex number|z|2=zz¯and when z is a real numberz=z¯

Q: Let C be a fixed nonzero complex number and note that i is a multiple valued, what restriction must…

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Q: Find all non - zero complex numbers z satisfying z =i z.

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Q: For any complex numbers z1 and z2 prove that |z1z2| = |z1||z2|.

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Q: For all complex numbers Zq,Z2 prove that |z, +z2| <]z|+]z2l

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Q: Give an example of a reduced cubic in the form x = 3px + 2q with three real roots but where q – p³ <…

A: Question: Give an example of a reduced cubic in the form x3=3px+2q with three real roots but where…

Q: Find the exact solutions to ​f(x)equals 0 in the complex numbers and confirm that the solutions are…

A: Given that f(x)=0 y=f(x) x2+4x+40=0

Q: If z is a complex number that satisfies the condition: |z − 2 − 3i| = 2, then show that the…

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Q: 5. Find the 3 3rd roots of -27i, and graph them in the complex plane.

A: -27i=27cos-π2+isin-π2 Now, use De Moivre's Formula

Q: For every real number a > 0, Find all complex numbers z, sati sfying the equati on z|-|+ a +i= 0.

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Q: Find all complex z that satisfy z^6 = -1

A: See the attachment

Q: The modulus of a complex number x+jy is equal to

A: The modulus of a complex number x+jy is equal to

Q: Find all complex solutions of 3x-x+4 0. (If there is more than one solution, separate them with…

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Q: If and are two complex numbers and and 2 are their conjugates respectively, then prove that + is a…

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Q: Derive the quadratic formula for the zeros of ax2+ bx + c, where a, b, and c are complex numbers and…

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Q: Find all complex numbers z satisfying the equation 2|=|* + =' -5+iV3 =0.

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Q: Find all of the complex third roots of i .

A: given complex number is i. we have to find all of the complex third roots.

Q: Solve the following inequality in the complex variable z: Re ((z − 1)(z − 2i)) ≥ Re(z − 1) · Re(z −…

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Q: Find all complex solutions of 2x + 3x+ 6 = 0. (If there is more than one solution, separate them…

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Q: Find all complex numbers z satisfying = -i

A: i = -1(i)2 = -1

Q: For all complex numbers Zq,Z2 prove that |Z,+z2| <]z¡\+]z2l

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Q: Find all complex numbers z of the form 2 =x+iy such that that ZZ = = 25 and *+ y = 7.

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Q: 5. Find all complex numbers of the form z = a + bi , where a and b are real numbers such that zz' =…

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Q: For all complex numbers Z1,Z, prove that |2z|+|*z|s]°z+z[

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Q: The set of all complex numbers z such that |z – 1| 2 is

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Q: Find all complex fourth roots of -2-2/3 i. In other words, find all complex solutions of x" =-2-2/3…

A: Given  -2 - 2√3i and we have to find all four roots of complex

Q: If the complex number z and its conjugate z' satisfy zz' + 2(z - z') = 12 + 8i, then the values of z…

A: Explanation of the answer is as follows