Solution of a Quadratic Equation by Completing the Square

In this section, we shall study Solution of a Quadratic Equation by Completing the Square,

Consider the following situation:
The product of Sunita’s age (in years) two years ago and her age four years
from now is one more than twice her present age. What is her present age?
To answer this, let her present age (in years) be x. Then the product of her ages
two years ago and four years from now is (x – 2)(x + 4).

Therefore, (x – 2)(x + 4) = 2x + 1
i.e., x2 + 2x – 8 = 2x + 1
i.e., x2 – 9 = 0

So, Sunita’s present age satisfies the quadratic equation x2 – 9 = 0.
We can write this as x2 = 9. Taking square roots, we get x = 3 or x = – 3. Since
the age is a positive number, x = 3.
So, Sunita’s present age is 3 years.
Now consider the quadratic equation (x + 2)2 – 9 = 0. To solve it, we can write
it as (x + 2)2 = 9. Taking square roots, we get x + 2 = 3 or x + 2 = – 3.
Therefore, x = 1 or x = –5
So, the roots of the equation (x + 2)2 – 9 = 0 are 1 and – 5.
In both the examples above, the term containing x is completely inside a square,
and we found the roots easily by taking the square roots. But, what happens if we are
asked to solve the equation x2 + 4x – 5 = 0? We would probably apply factorisation to
do so, unless we realise (somehow!) that x2 + 4x – 5 = (x + 2)2 – 9.
So, solving x2 + 4x – 5 = 0 is equivalent to solving (x + 2)2 – 9 = 0, which we have
seen is very quick to do. In fact, we can convert any quadratic equation to the form
(x + a)2 – b2 = 0 and then we can easily find its roots. Let us see if this is possible.
Look at Fig. 4.2.
In this figure, we can see how x2 + 4x is being converted to (x + 2)2 – 4.

Mathematical Modelling

First let us see why mathematical modelling is important,
Students are aware of the solution of word problems in arithmetic, algebra, trigonometry
and linear programming etc. Sometimes we solve the problems without going into the
physical insight of the situational problems. Situational problems need physical insight
that is introduction of physical laws and some symbols to compare the mathematical
results obtained with practical values. To solve many problems faced by us, we need a
technique and this is what is known as mathematical modelling. Let us consider the
following problems:
(i) To find the width of a river (particularly, when it is difficult to cross the river).
(ii) To find the optimal angle in case of shot-put (by considering the variables
such as : the height of the thrower, resistance of the media, acceleration due to
gravity etc.).
(iii) To find the height of a tower (particularly, when it is not possible to reach the top
of the tower).
(iv) To find the temperature at the surface of the Sun.
Appendix 2
(v) Why heart patients are not allowed to use lift? (without knowing the physiology
of a human being).
(vi) To find the mass of the Earth.
(vii) Estimate the yield of pulses in India from the standing crops (a person is not
allowed to cut all of it).
(viii) Find the volume of blood inside the body of a person (a person is not allowed to
bleed completely).
(ix) Estimate the population of India in the year 2020 (a person is not allowed to wait
till then).
All of these problems can be solved and infact have been solved with the help of
Mathematics using mathematical modelling. In fact, you might have studied the methods
for solving some of them in the present textbook itself. However, it will be instructive if
you first try to solve them yourself and that too without the help of Mathematics, if
possible, you will then appreciate the power of Mathematics and the need for
mathematical modelling.

Binary Operations

Let me explain you what is meant by Binary Operations,

Right from the school days, you must have come across four fundamental operations
namely addition, subtraction, multiplication and division. The main feature of these operations is that given any two numbers a and b, we associate another number a + b or a – b or ab or
, b ≠ 0. It is to be noted that only two numbers can be added or multiplied at a time.


When we need to add three numbers, we first add two numbers and the result is then added to the third number. Thus, addition, multiplication, subtraction and division are examples of binary operation, as ‘binary’ means two. If we want to have a general definition which can cover all these four operations, then the set of numbers is to be replaced by an arbitrary set X and then general binary operation is nothing but association of any pair of elements a, b from X to another element of X.

This gives rise to a general definition as follows:
Definition 10 A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote
∗ (a, b) by a ∗ b.

Statistics

Statistics

Statistics is the science of making effective use of numerical relating to groups of individuals or experiments.
Statistics is an attractive way of putting numbers across,statistics is about collection of information and its presentation and about drawing inferences from these. We come across facts and figures in the newspapers, Television and the radio. The numerical figures are called "the data".

Here are few terms that are most commonly linked to statistics:

A list of some important terms as follows: (i) ungrouped data (ii) tabulation of data (iii) range (iv) frequency (v) frequency distribution (vi) tally (vii) inclusive type of grouped frequency distribution (viii) exclusive type of grouped frequency distribution (ix) lower limit and actual lower limit (x) upper limit and actual upper limit (xi) class size or class width (xii) class mark or class mid-interval (xiii) Variables (xiv) Continuous Variables (xv) Discrete Variables.

Probability involves a lot of graphical representation of data.Figures are the most commonly used modes of graphical representation.The different types of graphs that we are going to study are 1. Bar graphs 2. Pie charts 3. Frequency polygon 4. Histogram.

Bar Graph

This is the simplest type of graphical presentation of data. The following types of bar graphs are possible: (a) Simple bar graph (b) Double bar graph (c) Divided bar graph.

Pie Graph or Pie Chart

Sometimes a circle is used to represent a given data. The various parts of it are proportionally represented by sectors of the circle. Then the graph is called a Pie Graph or Pie Chart.

Frequency Polygon

In a frequency distribution, the mid-value of each class is obtained. Then on the graph paper, the frequency is plotted against the corresponding mid-value. These points are joined by straight lines. These straight lines may be extended in both directions to meet the X - axis to form a polygon.

Histogram

A two dimensional frequency density diagram is called a histogram. A histogram is a diagram which represents the class interval and frequency in the form of a rectangle.
What do we mean by a Graphical representation of Data???

Graphs and diagram leave a lasting impression on the mind and make intelligible and easily understandable the salient features of the data. Forecasting also becomes easier with the help of graph. Thus it is of interest to study the graphical representation of data.

Triangles

Triangles

What is a Triangle??

A three sided closed figure is called a Triangle.

Triangles:

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.

Types of Triangles:

Classification of Triangles:

Triangles can be classified according to the relative lengths of their sides:

• In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
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• In an isosceles triangle, two sides are equal in length.
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• An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem. Some mathematicians define an isosceles triangles to have only two equal sides, whereas others define that an isosceles triangle is one with at least two equal sides.
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• The latter definition would make all equilateral triangles isosceles triangles
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• In a scalene triangle, all sides are unequal
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• The three angles are also all different in measure. Notice that a scalene triangle can be (but need not be) a right triangle.

Triangles are assumed to be two-dimensional plane figures.

A few basic theorems about similar triangles:

• If two corresponding internal angles of two triangles have the same measure, the triangles are similar.
• If two corresponding sides of two triangles are in proportion, and their included angles have the same measure, then the triangles are similar. (The included angle for any two sides of a polygon is the internal angle between those two sides.)
• If three corresponding sides of two triangles are in proportion, then the triangles are similar.

Two triangles that are congruent have exactly the same size and shape:all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)

Introduction to Trignometry

Introduction to Trigonometry:

Let us first learn how the word Trigonometry took Birth.

The word Trigonometry (from Ancient Sanskrit "trikona" or triangle and "mati" or measure or from Greek trigōnon "triangle" + metron "measure").

Trigonometry is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and angles in general, and the motion of waves such as sound and light waves.A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation.

History of Trignometry speaks about Ancient Egyptian and Babylonian mathematicians who lacked the concept of an angle measure, but they studied the ratios of the sides of similar triangles and discovered some properties of these ratios. The ancient Greeks transformed trigonometry into an ordered science.

Here is an example:

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

* The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.

* The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.

* The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.


Hope you like the above example of Trigonometry.
Please leave your comments, if you have any doubt

Polynomials

Polynomials

Introduction:

A Polynomial is an expression of finite length constructed from variables (also known as indeterminates).We can generally find polonomials in a wide variety and a wide range.For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science.

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) which may be multiplied by a finite number of variables (usually represented by letters). Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x¹, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant.

A polynomial is a sum of terms:

For example, the following is a polynomial:

Now lets learn about:Polynomial Functions:

Polynomial Functions:

A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

Next let us learn about "The Polynomial Equation"

Polynomial Equation:

A polynomial equation is an equation in which a polynomial is set equal to another polynomial.

is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x² – y², where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.