Equations Complex Numbers Quadratic Expressions Inequalities Absolute Value Equations & Inequalities Applications

Calculus

Billaaal

Contents

• Equations

• Complex Numbers

• Quadratic Expressions

• Inequalities

• Absolute Value Equations & Inequalities

• Applications

Linear equations– Linear Equation in One Variable

– Eq. does not have product of Two or more variables

Examples

• x2 + 5x -3 = 0

• 5 = 2x

• 5 = 2/x

• 3 – s = ¼

• 3 – t2 = ¼

• 50 = ¼ r2

• xy + x = 5

0 ,0 abax

Examples

• Linear equation in one variable

Example 1: 3x + 17 = 2x – 2. Find x.

• Linear equation in two or more variable

Example 2 : The eq. x = (y – b)/m has 4 variables.

Make y as a subject of Equation (Express Eq. in terms of y)

Word Problem(Very Important: Tagging quantities with variables)

Example:At a meeting of the local computer user group, each member brought two nonmembers. If a total of 27 people attended, how many were members and how many were nonmembers?

Solution:

• Let x = no. of members, 2x = no. of nonmembers

There were 9 members and 18 non

2 27

3 279 & 2 18

3

members at the meeti

39

ng.

x x

xx

Solve problems involving consecutive integers.

Consecutive integers: Two integers that differ by 1. e.g.. 3 and 4.

In General:

x= an integer,

x+1= next greater consecutive integer.

Consecutive even integers: such as 8 and 10, differ by

2.

Consecutive odd integers: such as 9 and 11, also differ

by 2.

Example:

Two pages that face each other have 569 as the sum of their page numbers. What are the page numbers?

Solution:

• Let x = the lesser page no.

• Then x + 1= the greater

page no.

• The lesser page number is 284, and the greater page number is 285.

1 569

2 5682 1 569

284, 1 285

1 12 2

284

x x

xx

x

Do by yourself :

1. Two consecutive even integers such that six

times the lesser added to the greater gives a sum of

86. Find integers.

2. The length of each side of a square is

increased by 3 cm, the perimeter of the new

square is 40 cm more than twice the length of

each side of the original square. Find

dimensions of the original square.

3. If 5 is added to the product of 9 and a

number, the result is 19 less than the number.

Find the number.

Complex numbers

General form of a Complex Number:

a+bi,

• a and b are reals

• i is an imaginary number.

What is an imaginary number?

A number for when squared gives – 1

Real and Complex part of a Complex Number: a+bi

Algebraic Operations on Complex Numbers

• Adding or subtracting complex number,

[Combine like terms].

Example: Simplify (8 – 3i) + (2+25i) - (12 – 3i)

• Multiplying complex number

Example: Solve: – 4i . 7i

Example: Simplify (3 –2i) (4+5i)

• Division in Complex Numbers

Complex Conjugate (Examples)

Example: Write the complex number

in Standard form: 8i / (6-5i)

Quadratic equation

Second degree polynomial equation in ONE Variable

General form:

Example: Using the zero factor property

Solve:

2 0a x b x c

2 5 6x x

Square Root Property

Very Important:

Example: Solve x2 = 49

Method of Completing Square

Example: Solve y2 - 2y = 3

Solution:

y2 - 2y + 1 = 3 + 1

(y + 1)2 = 4

y = 3 or 1

2x x

2 2x k x k x k x k

Quadratic Formula• Do you know how do we get to Quadratic Formula:

Example: Solve the Eq. by using Quadratic Formula: 3y2 + 9y = 2

Benefit of QUADRATIC FORMULA?

Discriminant: D =b2 – 4ac

It tells about Nature of the Roots:

Method: Check if D = 0, >0 or <0

Example: Discuss the nature of the roots of 2x2 +7x – 11 = 0

Solution:

22 4

02

b b acax bx c x

a

2Discriminant: [2 Real, Distinct7 4(2)( 1 & Irra1) 137 tional]

Equations Reducible to Quadratic Equations

1. 1. Equation with Rational Expression

1. 2. Equations with Radical Signs

2. 3. Equations with Fractional Powers

3. 4. Equations with integer powers

2

5 31

4x x

5 2 1x x

2/3 1/3

2/5

( 1) ( 1) 2

( 3) 4

x x

x

4 216 65 4 0x x

Inequalities

• Inequality Signs

• Rules of Algebraic Operations

• Linear Inequalities

• Quadratic Inequalities

• Absolute Value Equations

• Absolute Value Inequalities

• Compound Inequalities

Examples

Linear Inequality: 4q+3 < 2(q+3)

•

Quadratic Inequalities:

•

Absolute Value Equations:

Absolute Value Inequalities:

Compound Inequalities:

Rational Inequality:

•

2 3 4 0 ;

Ans : ( , 1] [4, ).

x x

44 2; 7

2

xx

x

4 2 5 8x

44 2; 7

2

xx

x

21

x

• Example: River Cruise

A 3 hour river cruise goes 15 km upstream and then

back again. The river has a current of 2 km an hour.

What is the boat's speed and how long was the

upstream journey?

Hints:

• Let x = the boat's speed in the water (km/h)

• Let v = the speed relative to the land (km/h)

going upstream, v = x-2

going downstream, v = x+2

Answer: x = -0.39 or 10.39

Time = distance / speedtotal time = time upstream + time downstream

• Boat's Speed = 10.39 km/h

• upstream journey = 15 / (10.39-2) = 1.79 hours = 1 hour 47min

• downstream journey = 15 / (10.39+2) = 1.21 hours = 1 hour 13min

Question:The current in a river moves at 2mph. a boat travels 18 mph upstream and 7mph down stream in a total 7 hours. what the speed of the boat in still water?

Ans: x = 3.8 mph in still water

Example: Two Resistors In Parallel

Total resistance is 2 Ohms, and one of the resistors is known to be 3 ohms more than the other.

Find: Values of the two resistors?

Solution:

Formula:

• R1 cannot be negative, so R1 = 3 Ohms is the answer.

• The two resistors are 3 ohms and 6 ohms.

2 1

1 2

2

1 1

1 1

1 1

1 1 1, 2, 3

1 1 16 0

2 3

2 3

T

T

R R RR R R

R RR R

R or R

Others

• Quadratic Equations are

useful in many other areas:

• For a parabolic mirror,

a reflecting telescope or

a satellite dish, the shape is defined by a quadratic equation.

• Quadratic equations are also needed when studying lenses and curved mirrors.

• And many questions involving time, distance and speed need quadratic equations.