In this lecture 1 we are going to cover : Equations Complex Numbers Quadratic Expressions Inequalities Absolute Value Equations & Inequalities Applications

1. Calculus
Instructor: Faiqa Ali
Student :Badar Shahzad

2. Contents
•Equations
•Complex Numbers
•Quadratic Expressions
•Inequalities
•Absolute Value Equations & Inequalities
•Applications

3. Equations
Key Operation: To Solve an
Equation?

4. 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

5. 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)

6. 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

7. 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.

8. 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.

9. 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.

10. 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

11. Algebraic Operations on Complex Numbers

12. •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)

13. Quadratic equation
Second degree polynomial equation in ONE Variable
General form:
Example: Using the zero factor property
Solve:

14. 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

15. 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:

16. Equations Reducible to Quadratic
Equations
1.1. Equation with Rational Expression
1.2. Equations with Radical Signs
1.3. Equations with Fractional Powers
1.4. Equations with integer powers

17. Inequalities
•Inequality Signs
•Rules of Algebraic Operations
•Linear Inequalities
•Quadratic Inequalities
•Absolute Value Equations
•Absolute Value Inequalities
•Compound Inequalities

18. Examples
Linear Inequality: 4q+3 < 2(q+3)
•
 Quadratic Inequalities:
•
Absolute Value Equations:
Absolute Value Inequalities:
 Compound Inequalities:
Rational Inequality:

19. •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 / speed
total time = time upstream + time downstream

20. •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

21. 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.

22. 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.