Rational Equations Solving Online

Learning equation of the form P(x)/Q(x) over the set of real numbers and Q(x) ≠ 0 where P(x) and Q(x) are two polynomials is called rational equation. Rational equations made easy for learning through online.

Example for rational equations: 3/8 = 3/(d + 4)

(x4 + x3 + x + 1)/(x + 5) = 9/5

1/(x + 9) = 3

12a-6 = 3a-2

Example problem for rational equations solving online:


The following example problems give idea for solving rational equations.

Example 1:

Find x value of the rational equation (6x + 36) / (9x + 54) = 6x

Solution:

Step 1: Given equation

(6x + 36) / (9x + 54) = 6x

Step 2: Take 6 and 9 out as common term from numerator and denominator respectively.

6(x + 6) / 9(x + 6) = 6x

Step 3: Cancel the term (x + 6)

6/9 = 6x

Step 4: Rearrange the above equation,

6/(6 * 9) = x

1/9 = x

x = 1/9

Example 2:

Find out x value for the equation x/(x - 4) + (1/x - 8) = 6/(x2 - 12x + 32)

Solution:

Step 1: Given equation

x/(x - 4) + (1/x - 8) = 6/(x2 - 12x + 32)

Step 2: Replace the term (x2 - 12x + 32) by (x - 4) (x - 8)

x/(x - 4) + (1/x - 8) = 6/ (x - 4) (x - 8)

Step 3: Make common denominator.

(x/(x - 4))((x - 8)/(x - 8)) + ((1/x - 8))((x - 4)/(x -4)) = 6/ (x - 4) (x - 8)

(x2 - 8x) / (x - 4)(x - 8) + (x - 4) / (x - 4)(x - 8) =  6/ (x - 4) (x - 8)

Step 4: Cancel the term (x - 4)(x - 8), we get

(x2 - 8x) + (x - 4) = 6

Step 5: Rearrange the above equation

x2 - 7x - 4 = 6

x2 - 7x - 4 - 6 = 0

x2 - 7x - 10 = 0

Step 6: On factorizing, we get

x = 8.217, - 1.217

This is how the rational equations can be solved through online.


Homework problem for rational equations solving online:


A few homework problems are given below  for solving rational equations.

1) Simplify and find x for the equation (4x + 12)/(4x + 32) = 1/(x + 8)

2) Find x value for the equation 1/(x + 7) = 4

3) Solve and find x for the equation (x2 - 12x + 32)/(x - 8) = 8

Solutions:

1) x = -2

2) x = -5

3) x = 12

Improper Integrals Solve Online

In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.

Specifically, an improper integral is a limit of the form


In which one takes a limit in one or the other (or sometimes both) endpoints. (Source: Wikipedia)


Example problems for solve online improper integrals


Online improper integrals example 1:

Solve:

Find the value of the integration function

`lim_(b->0) int_3^b(2x + 3)dx`

Solution:

Integrate the given function with respect to x, we get

= `lim_(b->0)` [2 `(x^2 / 2)` + 3x]b3

Substitute the lower and upper limits, we get

=  `lim_(b->0)` (b2 - 3b) - (9 + 9)

Substituting the value of b, we get

= (0) - 18

After substituting the limits, we get

= - 18

Answer:

The final answer is - 18

Online improper integrals example 2:

Solve:

Find the value of the integration function

`lim_(b->2) int_0^b(7x^3 + 3x^2)dx`

Solution:

Integrate the given function with respect to x, we get

= `lim_(b->2)` [7 `(x^4 / 4)` + 3`(x^3 / 3)` ]b0

Substitute the lower and upper limits, we get

=  `lim_(b->2)` [`(7 / 4)` b4 + b3] - (0)

Substituting the value of b, we get

= [(7 / 4) 24 + 23] - 0

After substituting the limits, we get

= 36

Answer:

The final answer is 36

Online improper integrals example 3:

Solve:

Find the value of the integration function

`lim_(b->1) int_2^b(6x^2 + 18x)dx`

Solution:
Integrate the given function with respect to x, we get

= `lim_(b->1)` [6 `(x^3 / 3)` + 18 `(x^2 / 2)` ]b2

Substitute the lower and upper limits, we get

=  `lim_(b->1)` (2b3 + 9b2) - (16 + 36)

Substituting the value of b, we get

= (11) - 52

After substituting the limits, we get

= - 41

Answer:

The final answer is - 41


Practice problems for solve online improper integrals


Online improper integrals example 1:

Solve:

Find the value of integration of the function

`lim_(b->0) int_5^b(6x)dx`

Answer:

The final answer is - 75

Online improper integrals example 2:

Solve:

Find the value of integration of the function

`lim_(b->6) int_0^b(12x + 2)dx`

Answer:

The final answer is 228

Mean Statistics Definition

Introduction:

Let us see the introduction of mean statistics. In statistics the mean is a mathematical average of a set of numbers. The average of the mean statistics is dividing the total by the number of scores and calculated by adding up two or more scores. Mean Statistics like as many other sciences of a developing discipline. In statistics has been defined in different times and different manners. We discuss the definitions of mean statistics.


Mean Definition:

The definition of the mean in mathematics, the statistical discrete random variable average of all the terms. A finite set of terms are forms rarely used in statistics are other expressions for the mean. The lowercase Greek letter mu (µ) is the expected value. The average value of mean statistics is a numerical set. The number of members in the group of numbers is by dividing the sum of a set of numbers.

Statistics Definition:

Statistics is definition of the technique or the scientific method is used to analyzing, collecting, interpreting, classifying, and data. The statistics is used to obtain the analyses, summaries, compare and present the numerical data. The statistics definitions are a science which deals with the application and investigates the statistical significance. 

These are the definitions of mean statistics.

Examples:

1. Find the mean of the following numbers 2, 5, 23, 15, 15, and 6.

Solution:

The given numbers are 2, 5, 23, 15, 15, and 6.

The average of set of numbers =2+5+23+15+15+6 / 6

= 36 / 6

=6

Answer: Mean is 6

2. Find the mean of the following numbers 5, 7, 34, 56, 23, 46, 56, and 12.

Solution:

The given numbers are 5, 7, 34, 56, 23, 46, 56, and 12.

The average of set of numbers =5+7+34+56+23+46+56+12/8

=   239 / 8

=29.88

Answer:

Mean is 29.8

3. Find the mean of the following numbers 56, 34, 23,  and 15.

Solution:

The given numbers are 56, 34, 23,  and 15.

The average of set of numbers=56+34+23+15/4

= 128 / 4

=32

Answer: Mean is 32

These are the example problems for definition of mean.

Number line

Learn on number line in this page and gain quality algebra help. First a brief introduction is given on the whole concept of number line and further the topic is explained.

Mathematically, a number line is referred to as an image of a straight line, which has several points on it as real numbers. Those points are indicated by integers. Particularly marked points with even spaces expose them on the line. Thus, real numbers are represented in each direction. Number line has both positive number and negative number held at correct points on line, in that zero is the center point of the number line, while right side of zero is positive number and left side of zero is negative number.



Numbers on a line can be represented horizontally as well as vertically. Normally, number line is represented horizontally.


Steps to draw


The following  are the steps to draw the number line-

• Step 1: Draw a horizontal straight line. Because mostly the number line is represented as horizontal line
• Step 2: Draw the arrow on both ends of number line.
• Step 3: Point the origin zero on the number line.
• Step 4: Write positive integer on the right side of origin with even spaces.
• Step 5: Write negative integer on the left side of origin with even spaces.
• Step 6: Mark all integers over the number line.
• Step 7: Plot the answers for given question.

Points on a number line


Ex 1: There are three persons on the origin namely as A, B, C. A walks backwards 2 points; B & C walks towards 4 points and 3 points respectively. Point out their current position.

Sol:




Points on a number line in decimal form

Case 1: Positive Decimal on number line

Let take the positive decimal as 3.75 and points it as in number line.

Generally, we know that decimal have two section,

• before the decimal point
• after the decimal point.

When make the decimal number on number line,

Step 1: keep the number before the decimal point. Here the 3 is the positive number before the decimal point.

Step 2: Now after the positive number three on a number line, we count the number in number line in between 3 and four as per the number have after the decimal point.

Step 3: Now we get the correct points for the given decimal as 3.75

Case 2: Negative decimal number on simple number line

Let take the number as -6.5 and mark it on the number line.



Step 1: keep the number before the decimal point. Here the 6 is the negative number before the decimal point.

Step 2: Now after the negative number 6 on a number line, we count the negative number in number line in between 6 and 7 as per the number have after the decimal point.

Random Occurrence

Let us see about the experiment of random occurrence,

              An outcome is produced from an operation through experiment is called as random experiment and also produce different possible outcomes. In a random experiment, an outcome random experiment is unpredictable.

Some more examples for random occurrence in random experiment:

• Coin is Tossing
• Die is rolling
• Take a card from a packet of card.
• Take a possible ball from bag contains different balls.

Some definitions about random occurrence:

Let we see about some definitions about random occurrence,

Trial: Random experiment performing by this. 

Sample space: In a random occurrence,the possible outcomes are taken by in set is known as sample space and is represented as S. When we roll a die, the possible outcomes are 1, 2, 3, 4, 5, 6 .
Sample space is S = { 1, 2, 3, 4, 5, 6} 

Event: Possible outcome or combination of outcomes is known as an event.An each subset of the sample space S is known as an event. Events represented as  A, B, C, D, E. While coin is tossing, getting a head or tail is take as an event. S = { H, T}, A = {H}, B = {T}.

Formula for random occurrence:

Let we see about formula for random occurrence,      

If a sample space contains n outcomes, m of which are favorable to an event E, then the probability of an event E, denoted by P (E), 
Number of favorable outcomes
  P(E)  =  Total number of outcomes number of favorable outcomes / Total number of outcomes

           = P(E) / P(S)

here,P(E) - Total number of outcomes number of favorable outcomes.

       P(S) - Total number of outcomes.

Examples:

1) Find the random occurrence (probability) to getting two heads when two coins are tossed simultaneously?

Solution:

In tossing two coins the sample space S = {HH, HT, TH, TT}, n(S) = 4.
Let A denote the event of getting two heads A = {HH}, n(A) = 1.
Therefore,probability to getting two heads P =n(A)/n(S) = 1/4.

2) Find the random occurrence getting 3 when rolling a die?

Solution:

 In rolling a die, the sample space S ={ 1, 2, 3, 4, 5, 6} : n (S) = 6.
 Let A be an event of getting 3
 A = { 3 }, n (A) = 1

∴ P(A)  = n(A) / n(S)

= 1/ 6.

Logarithms Learning

Logarithms function is the inverse of exponential function. If f(x)=logb(x) where b is the base of the logarithm function ,b>0and b is not equal to one .The power to which is base ten and base e(e=2.5466) obtain raised in number. The logarithm of a power equals the exponent multiplied with the logarithm of the base.

Examples:

1. f(x) = log2x , base is 2

2. g(x) = log4x, base is 4

3. h(x) = log0.5x, base is 5


Properties of logarithms Learning:


1.logbmn  =  logbm + logbn

The logarithm of a product is equal to the sum of the logarithms of each factor."

2. logb m/n  = logbm − logbn

The logarithm of a quotient is equal to logarithm of the numerator minus the logarithm of the denominator.

3. logb yn = n logby

The logarithm of a power of y equal to exponent of that power times the logarithm of y.

Some basic property of logarithms

loga1=0

logaa=1

logaxa=alogax


Change of base Rule for logarithms Learning:


By the definition of the base two logarithm

log2x = log2x

=> x = 2log2x ( as log2x = a then x = 2a)

(since immediate consequences of the definition of logarithms ) that the logarithm of a power equals the exponent multiplied with the logarithm of the base. Therefore by taking natural logarithm on both sides of the proceeding equation, obtain

ln(x) = log2(x)ln(2)

(since, immediate consequences of the definition of logarithms ) that the logarithm of a power equals the exponent multiplied with the logarithm of the base. Therefore by taking natural logarithm on both sides of the preceding equation obtains

Solving for base two logarithm gives the same formula as before:

log2(x)=`(ln(x))/(ln(2))`


Examples of logarithms Learning:


Example1:Solve log23+log28=log2 (4x)

Solution: logarithmic function log23+log28=log2 (4x)

log2(3*8)  = log2(4x)

log2(4x)=log2(24) by property of logarithm (1)

Equate both sides, s the bases are same 2,  4x = 24

Simplification: x = 24/4

Answer = 6

Example2: log654-log69

Solution:

lo654-log69=log6(54/9) by property of logarithm (2)

simplification: log66

Answer = 1

Examble3: logarithm Solving equation in    log284

Solution:Converting the logarithmic equation

=  4log28  by property logarithm (3)

Simplification: 4log223 = 12log22

Answer=12

Project on Trigonometry for Class 10

The word "Trigonometry" is derived from three Greek words----' tri ' means three, 'gonia' means angle, and 'metron' means measure.Thus "Trigonometry" means  "three angle measure". Trigonometry deals with the relation between the angles and sides in a triangle.The study of trigonometry is of great importance in several fields and applied in many branches of  Science and Engineering such as Seismology,design of electrical circuits,estimating the heights of tides in the ocean etc.The three main trigonometric functions are sine,cosine and tangent and their reciprocals are co secant,secant and cotangent respectively.

Definitions of Trigonometric functions:

Definitions of Trigonometric Functions :-

Consider a right angle triangle and Let P(x,y) be the point and θ be the acute angle.

ON = x ; NP = y ; OP = r



The sine function is defined as ratio of opposite side (y) to hypotenuse(r).
sinθ = `(y)/(r)`

The cosine function is defined as ratio of adjacent side (x) to hypotenuse(r).
cosθ = `(x)/(r)`

The Tangent  function is defined as ratio of opposite side (y) to adjacent side (x).
tanθ = `(y)/(x)`

The reciprocal of sine is co secant and is defined as
cscθ = `(r)/(y)`

The reciprocal of cosine is secant and is defined as
secθ = `(r)/(x)`

The reciprocal of tangent is cotangent and is defined as
cotθ = x/y

Note:

sinθ and cscθ are reciprocal => sinθ * cscθ = 1
cosθand secθ are reciprocal => cosθ * secθ = 1
tanθ and cotθ are reciprocal => tanθ *  cotθ = 1


Formulas and Identities of Trigonometric functions


Identities :

sin2θ + cos2θ = 1
1 + tan2 θ = sec2 θ
1+ cot2 θ = csc2 θ

Formulas

sin(A+B) = sinA cosB + cosA sinB
sin(A-B) = sinA cosB - cosA sinB
cos(A+B) = cosA cosB - sinA sinB
cos(A-B) = cosA cosB + sinA sinB
tan(A+B) = `(tan A + tan B)/(1- tan A tan B)`
tan(A-B) = `(tan A - tan B)/(1+ tan A tan B)`

Signs of Trigonometric Functions:

The entire coordinate plane is divided into 4 quadrants and they are named in counter clockwise direction.Let P(x , y) be a point in the coordinate plane.



(1) If P lies in the 1st quadrant,  all the trigonometric functions are positive.

(2) If P lies in the 2nd quadrant,  sinθ, cscθ are positive and the others are negative.

(3) If P lies in the 3rd quadrant, tanθ, cotθ are positive and the others are negative.

(4) If P lies in the 4th quadrant, cosθ, secθ are positive and the others are negative.\