Monday, April 8, 2013

Quadratic Inequalities


In this page we are going to discuss about  quadratic inequalities concept.A quadratic inequality in one variable is in the form of an expression:

ax2 + bx + c ≤ 0  or   ax2 + bx + c < 0
ax2 + bx + c ≥ 0  or   ax2 + bx + c < 0


where a, b and c are real numbers, a ≠ 0. The values of x which satisfy the given inequality are called the solutions of the inequality.

A quadratic equation has only two roots. But a quadratic inequality has many roots.


Methods to solve quadratic Inequalities


There are two methods to solve quadratic inequalities -

Method 1: Finding the solution by dividing the given polynomial into factors. This method is called the 'Algebraic method'.

Method 2: Finding  the solutions by drawing the graph of the inequality. This method is called the 'Graphical method'.

Note:

  • The trick in solving a quadratic inequality is to replace the inequality symbol with an equal sign and solve the resulting equation.  The solutions to the equation will allow us to establish intervals that will let you solve the inequality.
  • Plot the solutions on number line creating the intervals for investigation. Pick any number from each interval and test it in original inequality. If the result is true, that interval is the solution to the inequality.

Solving quadratic inequalities


Below you can see the example on solving quadratic inequalities-

Example:1 Find the solution set of x2 – 4x – 21 ≥ 0 in

1. Algebraic Method
2. Graphical Method

Solution:

1.  Algebraic Method:

X2 - 4x - 21 ≥ 0
=> ( x + 3) ( x - 7 ) ≥ 0
Here the coefficient of x2 is > 0 and the sign of the quadratic expression is non-negative.
So, x ≥ -3 and x ≤ 7 or there are two cases for the product ( x + 3 ) ( x - 7 ) to be non-negative.

Case (1):
x + 3 ≥ 0 and x - 7 ≥ 0
=> x ≥ -3 and x ≥ 7
So it is necessary that  x≥ 7. Notice the line marked below with x ≥ -3 and x ≥ 7 and their intersection.

Quadratic inequalities

From the figure above, it is clear that the intersection is x ≥ 7.

Case (2):
x + 3 ≤ 0 and x - 7 ≤ 0
=>   x ≤ -3 and x ≤ 7
So it is clear x ≤ -3.
i.e., the values of x for which x2 - 4x - 21 ≥ 0 are given by these two cases.

All values of x satisfying the in equations x ≤ -3 or x ≥ 7 become the solution set for the given in equations. This solution set can  be shown in the graph given below.

Quadratic inequalities algebraic method

From the figure above, it is clear that the intersection is x ≥ 7.


2. Graphical Method:

The given polynomial function is f(x) = x2 - 4x - 21
x-3-2-101234
f(x)0-9-16-21-24-25-24-21



Using the above chart, we can draw a graph given below


Graphing quadratic inequalities


Now, we require the values of x satisfying x2 - 4x - 21 ≥ 0 , i.e., f(x) ≥ 0 ( y ≥ 0 ). That is we require the values of x for which f(x) lies on x-axis and above. But it is clearly understood from the figure that the values of  x are -3, left side to -3, +7, and right side to +7. Thus all values of x such that x ≤ 3 and x ≥ 7 become the solution set.

Friday, April 5, 2013

Review Geometry Learning


Geometry is learning a part of mathematics which deals with the dimensions of solids and surfaces, shapes, angles and lines etc., It reviews the deduction concepts and consequence logic's, which can be applied through out your life time . In this article, you can review the basic terms and shapes in geometry.

Learning Basics Terms in Geometry:

Learning the following definitions helps to review concepts in geometry.

Point:
A point have no dimensions, it just denotes a position.

Line:
A line can be defined as a straight curve which connects many points. A line has only one dimension, i.e., length.

Collinear points:
If two or more points lie on same line, then they are said to be as colinear points.

Plane:
A Plane can be defined as a set of points and line segments combined infinitely to form a flat surface, it can be extended infinitely in any directions. A plane has infinite length and width, but it doesn't have any height.

Midpoint:
A midpoint can be defined as a point, which divides a line segment equally from both the end points.

Line segment:
A line segment can be defined as a part of a line. It consists of two end points and it was represented by its end points.

Ray:
A ray can be defined as a line having one fixed end point and an infinite extension on the other end.

Angles:

An Angle can be formed by two rays having a common end point.These are the reviews of the basic terms in geometry learning.

Learning Basic Shapes in Geometry:


Square:
A Square can be defined as a regular quadrilateral with equal sides and 4 right angles.

Rectangle:
A Rectangle is a four sided geometrical figure with two pairs of same side lengths.

Triangle:
In geometry, only shape which has three sides and angles is a triangle.

Circle:
A circle is a geometric figure, formed by a locus of points which are all equidistant from a common point called as its center. The distance between the locus of points to the common point is known as radius of the circle.

These are the reviews of the basic shapes in geometry learning.

Wednesday, April 3, 2013

Long Integer Definition


Integer is a whole number which is not a fraction, that is it can be either positive, negative or zero. Thus the numbers -20, -10, 0, 10, 50, 98, 540 are integers. It can not have decimals. It is commonly used in computer programming as a data type. It can also be used to find out item's location in an array. If two integers are added, subtracted or multiplied, then the output is also an integer. Where as if one integer is divided with other integer, then the result may be an integer or a fraction.

When we think of the number, we first of all think of natural numbers: 1, 2, 3, 4, 5, ...... which are also called positive integers. By applying the operation of subtraction, we get the number zero and negative integers. The set {.....-4, -3, -2, -1, 0, 1, 2, 3, 4,........} is called the set of integers. By division operation, we get the positive and negative fractions. The integers and fractions together constitute the class of rational number p/q, where p and q are integers but q is not equal to 0.

Integers and long integers:


Numbers which includes positive integers{ 1, 2, 3, 4, .....} and negative integers{-1, -2, -3, -4,......} with zero{0} are called integers. There is no fractions or decimal points are included in the integers. They are represented by Z.

Therefore,

Z={ … -4, – 3, – 2, – 1, 0, + 1, + 2, + 3, +4 ….}

+ 1, + 2, + 3, … are positive integers.

– 1, – 2, – 3, … are negative integers.

Note:

Positive numbers can be written even without the ‘+’ sign.

For example, +5, +4, +3, +2, +1 are  written as 5, 4, 3, 2, 1.

Thus, Z = {…… – 3, – 2, – 1, 0, 1, 2, 3, ……}

The integers are represented on the number line as follows:

Negative number and Postivie number

On the number line, the numbers which are to the right of zero are called positive integers. The numbers which are to the left of zero are called negative integers.

(1)  Every natural number or a whole number.

(2)  Every whole number is an integer.

Order in long integer:

  • On the number line, we find that the number value increases as we move to the right and decreases as we move to the left.
  • If we represent two integers on the number line, the integer on the right is greater than the integer on the left.
  • In other words, the integer on the left is lesser than the integer on the right.
For example, consider the following points marked on the number line in the figure given below:

Number line
In the above figure,

4 is to the right of 2                `:.` 4 > 2

2 is to the right of -1               `:.` 2 > -1

-5 is to the left of  -1                 `:.` -5 < -1


Example problems:


Example 1: Which is smaller? -2 and -5

Solution:

First, mark the integers -2 and -5 on the number line.

number line(a)

On the number line, -5 is on the left side of -2.

Therefore, -5 is smaller than -2

That is -5 < -2.

Example 2: Write the following integers in ascending order.

3,-2,0,-4,-1,5

Solution:

First mark these integers on the number line.

number line(b)

Now arrange the integers from left to right to get them in ascending order

-4 < -2 < -1 < 0 < 3 < 5

Therefore, the ascending order is -4, -2, -1, 0, 3, 5.