COMPLEX NUMBERS AND COMPLEX NUMBERS AND QUADRATIC EQUATIONSQUADRATIC EQUATIONS

From: Deepanshu ChowdharyClass: 11th A

Complex Numbers a + bi

Complex numbers are numbers of theform a + bi, where a and b are realnumbers. The real number a is called thereal part of the number a + bi; the realnumber b is called the imaginary part ofa + bi.

(a + bi) + (c + di) = (a + c) + (b + d)i

(2 + 4i) + (-1 + 6i) = (2 - 1) + (4 + 6)i

= 1 + 10i

Sum of Complex Numbers

(a + bi) - (c + di) = (a - c) + (b - d)i

(3 + i) - (1 - 2i) = (3 - 1) + (1 - (-2))i

= 2 + 3i

Difference of Complex Numbers

Product of Complex Numbers

If z=a +bi is a complex number, then its conjugate, denoted by

TheoremThe product of a complex number and its conjugate is a nonnegative real number. Thus if z=a +bi, then

Theorem

If N is a positive real number, we define the principal square root of -N as

In the complex number system, the solution of the quadratic equation

where a, b, and c are real numbers and are given by the formula

Solve:

Discriminant of a Quadratic Discriminant of a Quadratic EquationEquation

is called a discriminant

>0, there are 2 unequal real solutions.

=0, there is a repeated real solution.

<0, there are two complex solutions. The solutions are conjugates of each other.

We can add, subtract, multiply or divide complex numbers. After performing these operations if we’ve simplified everything correctly we should always again get a complex number (although the real or imaginary parts may be zero). Below is an example of each to refresh your memory.

(3 – 2i) + (5 – 4i)ADDINGCombine real parts and combine imaginary parts= 8 – 6i

(3 – 2i) - (5 – 4i)SUBTRACTING

= -2 +2i

Be sure to distribute the negative through before combining real parts and imaginary parts3 – 2i - 5 + 4i

(3 – 2i) (5 – 4i)MULTIPLYING FOIL and then combine like terms. Remember i 2 = -1

= 15 – 12i – 10i+8i2

=15 – 22i +8(-1) = 7 – 22iNotice when I’m done simplifying that I only have two terms, a real term and an imaginary one. If I have more than that, I need to simplify more.

DIVIDING

ii

4523

Recall that to divide complex numbers, you multiply the top and bottom of the fraction by the conjugate of the bottom.

ii

4545

This means the same complex number, but with opposite sign on the imaginary term

FOIL2

2

162020258101215

iiiiii

12 i

116202025

18101215

iiii

Combine like terms

41223 i

We’ll put the 41 under each term so we can see the real part and the imaginary part

i412

4123

Let’s solve a couple of equations that have complex solutions to refresh our memories of how it works.

0252 x-25 -25

252 x

ix 25

01362 xxa

acbbx2

42

Square root and don’t forget the The negative

under the square root becomes i

Use the quadratic formula

12

131466 2 x

252366

2166

2166 i

246 i

i23

i5

Powers of i

12 iii

iiiii )(123

111224 iii iiiii 145

111246 iii iiiii 1347

111448 iii

We could continue but notice that they repeat every group of 4. For every i 4

it will = 1

To simplify higher powers of i then, we'll group all the i 4ths and see what is left.

iiiii 88433 1

4 will go into 33 8 times with 1 left.

iiiii 320320483 1

4 will go into 83 20 times with 3 left.

aacbbxcbxax

240

22

If we have a quadratic equation and are considering solutions from the complex number system, using the quadratic formula, one of three things can happen.

3. The "stuff" under the square root can be negative and we'd get two complex solutions that are conjugates of each other.The "stuff" under the square root is called the discriminant.

This "discriminates" or tells us what type of solutions we'll have.

1. The "stuff" under the square root can be positive and we'd get two unequal real solutions 04 if 2 acb2. The "stuff" under the square root can be zero and we'd get one solution (called a repeated or double root because it would factor into two equal factors, each giving us the same solution).04 if 2 acb

04 if 2 acb

The Discriminant acb 42

Solving

Equations

A quadratic equation is an equation equivalent to one of the form

Where a, b, and c are real numbers and a 0

02 cbxax

To solve a quadratic equation we get it in the form above and see if it will factor.

652 xx Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side.

-5x + 6 -5x + 6

0652 xx Factor.

023 xx Use the Null Factor law and set each

factor = 0 and solve.02or 03 xx 3x 2x

So if we have an equation in x and the highest power is 2, it is quadratic.

In this form we could have the case where b = 0.

02 cbxaxRemember standard form for a quadratic equation is:

02 cax002 cxax

When this is the case, we get the x2 alone and then square root both sides.

062 2 x Get x2 alone by adding 6 to both sides and then dividing both sides by 2

+ 6 + 6

62 2 x2 2

32 xNow take the square root of both sides remembering that you must consider both the positive and negative root.

3x Let's check: 0632

2 0632

2

066 066

Now take the square root of both sides remembering that you must consider both the positive and negative root.

02 cbxaxWhat if in standard form, c = 0?

002 bxaxWe could factor by pulling an x out of each term.

032 2 xx Factor out the common x

032 xx Use the Null Factor law and set each

factor = 0 and solve.

032or 0 xx

23or 0 xx If you put either of these values in for

x in the original equation you can see it makes a true statement.

02 cbxaxWhat are we going to do if we have non-zero values for a, b and c but can't factor the left hand side?

0362 xx This will not factor so we will complete the square and apply the square root method.First get the constant term on the other side by subtracting 3 from both sides.362 xx

___ 3___ 62 xx

We are now going to add a number to the left side so it will factor into a perfect square. This means that it will factor into two identical factors. If we add a number to one side of the equation, we need to add it to the other to keep the equation true.

Let's add 9. Right now we'll see that it works and then we'll look at how to find it.

9 9 69 62 xx

69 62 xx Now factor the left hand side.

633 xx

two identical factors

63 2 xThis can be written as:

Now we'll get rid of the square by square rooting both sides.

63 2 x Remember you need both the positive and negative root!

63 x Subtract 3 from both sides to get x alone.

63 xThese are the answers in exact form. We can put them in a calculator to get two approximate answers.

55.063 x 45.563 x

Okay---so this works to solve the equation but how did we know to add 9 to both sides?

___ 3___ 62 xx 9 9

633 xx We wanted the left hand side to factor into two identical factors.

When you FOIL, the outer terms and the inner terms need to be identical and need to add up to 6x.

+3 x

+3x

6 x

The last term in the original trinomial will then be the middle term's coefficient divided by 2 and squared since last term times last term will be (3)(3) or 32.

So to complete the square, the number to add to both sides is…

the middle term's coefficient divided by 2 and squared

Let's solve another one by completing the square.

02162 2 xx To complete the square we want the coefficient of the x2 term to be 1.

Divide everything by 20182 xx2 2 2 2

Since it doesn't factor get the constant on the other side ready to complete the square.

___1___82 xx

So what do we add to both sides?16

16 16

Factor the left hand side 15444 2 xxxSquare root both sides (remember )

154 2 x

154 x 154 xAdd 4 to both sides to get x alone

2

28

the middle term's coefficient divided by 2 and squared

By completing the square on a general quadratic equation in standard form we come up with what is called the quadratic formula. (Remember the song!! )

aacbbx

242

This formula can be used to solve any quadratic equation whether it factors or not. If it factors, it is generally easier to factor---but this formula would give you the solutions as well.

We solved this by completing the square but let's solve it using the quadratic formula

aacbbx

242

1

(1)(1)

6 6 (3)2

12366

Don't make a mistake with order of operations! Let's do the power and the multiplying first.

02 cbxax

0362 xx

212366

x2

246

626424

2626

2

632

There's a 2 in common in the terms of the numerator

63 These are the solutions we got when we completed the square on this problem.

NOTE: When using this formula if you've simplified under the radical and end up with a negative, there are no real solutions.(There are complex (imaginary) solutions, but that will be dealt with in year 12 Calculus).

SUMMARY OF SOLVING QUADRATIC EQUATIONS

Get the equation in standard form: 02 cbxax If there is no middle term (b = 0) then get the x2 alone and square root both sides (if you get a negative under the square root there are no real solutions).

If there is no constant term (c = 0) then factor out the common x and use the null factor law to solve (set each factor = 0).

If a, b and c are non-zero, see if you can factor and use the null factor law to solve.

If it doesn't factor or is hard to factor, use the quadratic formula to solve (if you get a negative under the square root there are no real solutions).

aacbbxcbxax

240

22

If we have a quadratic equation and are considering solutions from the real number system, using the quadratic formula, one of three things can happen.

3. The "stuff" under the square root can be negative and we'd get no real solutions.

The "stuff" under the square root is called the discriminant.

This "discriminates" or tells us what type of solutions we'll have.

1. The "stuff" under the square root can be positive and we'd get two unequal real solutions 04 if 2 acb2. The "stuff" under the square root can be zero and we'd get one solution (called a repeated or double root because it would factor into two equal factors, each giving us the same solution).

04 if 2 acb

04 if 2 acb

The Discriminant acb 42

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au

THANKS FOR THANKS FOR WATCHING!!!WATCHING!!!