Complex Analysis: Find 2 Square Roots, Solve Eqn, Form Triangle

In summary, the third side will be a complex number with the value (-1+i)/2. Its ratios will be 1/2 to 1 and 1 to -1.
  • #1
mtayab1994
584
0

Homework Statement



1- Find the two square roots of the complex number z=3+4i.

2a- Solve in ℂ the equations: (E): 4z^2-10iz-7-i=0

b- Let a and b be solutions to (E) such that: Re(a)<0 and the two points A and B plots/pictures of a and b. Show that b/a=1-i. Conclude that AOB is an equilateral triangle.

The Attempt at a Solution



1- After solving (p+qi)^2=3+4i i found that the solutions were either 2+i or -2-i.

2-a For the complex equation i found two complex roots: z1=(-3+6i)/8 and z2=(3+14i)/8.

b- So i took the two solutions that i found from the previous question and chose a=z1 and b=z2 and after computing i got a whole different answer. Is my work correct, if not some help would be very much appreciated.
 
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  • #2
mtayab1994 said:
2-a For the complex equation i found two complex roots: z1=(-3+6i)/8 and z2=(3+14i)/8.
I get a different result. Pls post your working.
 
  • #3
mtayab1994 said:

Homework Statement



1- Find the two square roots of the complex number z=3+4i.

2a- Solve in ℂ the equations: (E): 4z^2-10iz-7-i=0

b- Let a and b be solutions to (E) such that: Re(a)<0 and the two points A and B plots/pictures of a and b. Show that b/a=1-i. Conclude that AOB is an equilateral triangle.

The Attempt at a Solution



1- After solving (p+qi)^2=3+4i i found that the solutions were either 2+i or -2-i.

2-a For the complex equation i found two complex roots: z1=(-3+6i)/8 and z2=(3+14i)/8.
If z= (-3+ 6i)/8 then [itex]z^2= [(9- 36)- 2(18i)]/64= -27/64- (9/8)i[/itex] so [itex]4z^2- 10iz- 7- i= -27/4- (9/2)i+ (15/4)i+ 15/2- 7- i= (-27/4+ 15/2- 7)+ (15/4- 9/2- 1)i= (-27+ 30- 28)/4+ (15/4- 18/4- 4/4)i= -25/4- (7/4)i, NOT 0.

b- So i took the two solutions that i found from the previous question and chose a=z1 and b=z2 and after computing i got a whole different answer. Is my work correct, if not some help would be very much appreciated.
 
  • #4
Sorry I was wrong on the roots of the equation they are correct now i got:

z1=(1/2)+(3i/2) and z2=(-1/2)+i and that certainly gives 1-i when you take z2=a and z1=b.

But the conclusion i can't quite fathom, any help with that please.
 
  • #5
mtayab1994 said:
But the conclusion i can't quite fathom, any help with that please.
What will the third side look like as a complex number (in terms of a and b)? What will its ratios be to the other two?
 

Related to Complex Analysis: Find 2 Square Roots, Solve Eqn, Form Triangle

1. What is Complex Analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers, functions, and their properties. It is a powerful tool used in many areas of mathematics, physics, and engineering.

2. How do you find the square roots of a complex number?

To find the square roots of a complex number, we can use the formula z1,2 = ± √(r(cosθ + isinθ)), where r is the modulus of the complex number and θ is the argument. We can also use the polar form of the complex number and use the fact that the square roots of a complex number are the same distance from the origin, but at angles that differ by half the argument.

3. How do you solve equations involving complex numbers?

To solve equations involving complex numbers, we can use the same techniques as solving equations with real numbers, such as factoring, completing the square, or using the quadratic formula. However, we also need to consider the complex solutions and use properties of complex numbers, such as the conjugate, to simplify the equation.

4. How do you form a triangle with complex numbers?

To form a triangle with complex numbers, we can use the complex plane or Argand diagram. The complex numbers can represent the vertices of the triangle, and we can use the distance formula and trigonometry to find the side lengths and angles of the triangle.

5. What are the applications of complex analysis?

Complex analysis has many applications in various fields, such as physics, engineering, and economics. It is used to solve differential equations, model fluid flow, analyze electrical circuits, and understand the behavior of systems with non-real solutions. It also has applications in signal processing, cryptography, and quantum mechanics.

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