Reviewing Algebra II (Part Four): Manipulating Complex Numbers, Solving Complex Equations

 Fortunately, there are a number of previous blog posts which will be of use in solving complex or imaginary problems. These are given below:

 Taking Complex Roots (1)

Solution of Complex Roots of Unity Problem

Solutions to Complex Roots Problems

More Complex Roots

Obtaining Complex Conjugates

Brane Space: Introducing Complex Functions (1)

We can begin with a few practice exercises in manipulating complex expressions, the aim being to simplify each:

a) (5 + 2i) (3 + 4i) 

b) (2 + 6i)²

(c)Express:  (5 + 10i)/ (½   -   2i/3 ) in the form a + bi

The solutions follow in order:

 (a): (5 + 2i) (3 + 4i) =5 (3 + 4i) + 2i (3 + 4i) 

In which we are just using the distributive law.

Then:  15 + 20i + 6i +  8i² = 15 + 26i + 8(-1)

= 15 + 26i - 8  Or:   7 + 26i (Simplest form)  

 (b): (2 + 6i)²   =   (2 + 6i) (2 + 6i)

Using the distributive law again:

 2(2 + 6i)  + 6i(2 + 6i) =  4 + 12i + 12i +  36 i²

= 4 + 24i  + 36 (-1) =  -32 + 24i (Simplest form) 

(c)(5 + 10i)/ (½   -   2i/3 ) 

  Þ

(5 + 10i) (6)/ (½   -   2i/3 ) (6)  

= 30 + 6o i/ 3 - 4i

 Þ

30 + 6o i (3 + 4i) / 3 - 4i (3 + 4i)  =

90 +  120i  + 180 i  + 240 ² /  (3²   +  4²  ) =

-150 + 300i/ 25   =  -6  + 12i (Simplest form) 

Finally, consider solving a complex equation:

x²   +  12 =  6x  

for which we need to find the complex roots.

Solution:

Begin by writing the equation in standard form:

x²  - 6x  + 12 = 0

This is easily seen to be a quadratic equation for which we will need to use the quadratic formula:

The key aspect to bear in mind is that the discriminant (term inside the radical sign) will always be a negative term for a complex equation. In this case, given a =1, b = -6, c = 12 we have:

 b ²  - 4ac    =  36 - 48  =  -12

Substituting values for the full formula:

x =  6 +  Ö(36 - 48)/  2(1) 

=

6 +  Ö -12/  2  =  6  +   Ö (4)/Ö (3)/ Ö (-1)/  2 

Þ

6 +   2i Ö (3) /  2 =

 3 + i Ö (3) 

Selected problems:

1)  Write: (2 + 3i)/ (1 + 2i) in the form a + bi

2) Show the product of a complex number a + bi and its conjugate (a - bi) is a real number. (You can use the result from (1).

3) Find the product: (3 - 5i) (2 + i)

4)  Write in the form a + bi:

Solve each of the following:

5)   x²   +  32  = 0  for all roots  

6) x⁴   - 1  = 0

7) x²   - 4x  + 8 = 0

8) x²   + 10x + 29 = 0

9) 4x²   - 4x  + 5 = 0

10) 4x - 7  = x²