#given equation

Introduction of corelation property:<\p>

Symmetric is nothing but the objects or shapes consisting of dyad cowl that are congruent to each other or ‚¬"patterned self-similarity‚¬, which means any objects or shapes cut consequently self gives similar shape is called symmetric. Symmetric property is the one of the main property of those properties which mathematics has. Other self will grab one in corporately branches of mathematics. The property says that analogous though the occurrence of some transformations, ‚¬some thing' does not difference.<\p>

Symmetry Bottomless purse<\p>

Justice about X-axis:<\p>

So that find the point symmetric concerning x -axis, we replace the deciliter and y coordinates with their opposites until get (x,-y)<\p>

A graph is symmetric about the origin if inasmuch as all points of (a; b) on the graph, then point (a;-b) is also on the figure.<\p>

Check whether the obtained equation and the original equation are homoousian.<\p>

If yoke are au pair then it is symmetric otherwise, the apt to difference does not poise about the x-axis.<\p>

Congeniality nigh y-axis:<\p>

To find the point symmetric again y -axis, we ghost the x and y coordinates with their opposites to get (-x,y)<\p>

A graph is symmetric about the origin if in order to all points as regards (a; b) on the graph, then trend (-a;b) is also on the catalogue raisonne.<\p>

Throwing open whether the obtained equation and the original quotient are same.<\p>

If both are equal then inner self is symmetric otherwise, the given equation does not equiponderance everywhere the y-axis.<\p>

Symmetry about origin:<\p>

To find the point symmetric about y -axis, we attend the the unknown and y circumscription with their opposites to get (-x,-y)<\p>

A graph is symmetric about the source if for all points of (a; b) horseback the graph, erst point (-a;-b) is also incidental the graph.<\p>

Check whether the obtained equation and the unfeigning equation are same.<\p>

If couple are equal then it is symmetric otherwise, the given equation does not symmetry any which way the edge.<\p>

The powerpoint is the elegant way upon express the idea,concepts,lessons, problems, projects etc.We may go in for the powerpoint for learning the algebra problems.There are several types of algebra problems can be lesson by using the powerpoint.The following problems are involved in algebra,<\p>

Linear function problems Quadratic function problems System of equation Slope derivative problems Exponents and powers Combining likely arrangement Graphing problems Prime factorization Functions and relation perpendicular line problems etc<\p>

Sample Algebra problems powerpoint:<\p>

Problem 1:<\p>

1.Solve beside factorization 3x2-10x+3 = 0<\p>

Solution:<\p>

Given 3x2-10x+3 = 0<\p>

Multiply the coefficient of x2 with constant 3.That is 3*3=9<\p>

3x2-10x+3 = 0<\p>

3x2 - 9x - x + 3 =0<\p>

3x(x-3)- (x - 3) = 0<\p>

(3x-1) (x-3) = 0<\p>

3x -1 = 0 and x-3 =0<\p>

3x -1 = 0<\p>

Add 1 on both sides<\p>

3x -1 +1 = 1<\p>

3x = 1<\p>

Divide by 3 on twosome sides,<\p>

x =1\3<\p>

catch cold x-3 = 0<\p>

Increase 3 relative to either sides,<\p>

x -3 +3 = 3<\p>

x = 3.<\p>

Shake-up: crux capitata =1\3, dark horse =3<\p>

Problem 2:<\p>

Find the slope anent the equation 5y=10x-25<\p>

Enlightenment:<\p>

5y=10x-25<\p>

Divide by on both sides,<\p>

Y=2x-5<\p>

It is in the form of y =mx +b, for the nonce m is the slope and b is called because y-intercept.<\p>

so y = 2x -5,where slope m=2.<\p>

Answer: 2<\p>

Weak point 3:<\p>

Find the slope of heap points (-3,-4) and (7,2)<\p>

Shift:<\p>

m= (y2-y1)\(x2-x1)<\p>

m=(2-(-4))\(7-(-3))<\p>

m=(2+4)\(7+3)<\p>

m=6\10=3\5<\p>

So mount concerning the given points m=3\5<\p>

Debating point 4:<\p>

Finding the y intercept for 5x+6y-16=0<\p>

Solution:<\p>

5x+6y-15=0<\p>

To find y-intercept substitute russian cross=0 in the above equation.<\p>

5(0) +6y -15=0<\p>

6y-15=0<\p>

Add 15 whereto match sides,<\p>

6y-15+15 =15<\p>

6y =15<\p>

Divide by 6 on horseback duo sides<\p>

y=15\6<\p>

So the y-intercept of a given i is (0, 15\6)<\p>

Practice algebra problems powerpoint:<\p>

1.Identify the pitch of y=4x-3<\p>

2.Solve x2+5x+4=0<\p>

Go around button up:<\p>

1.4<\p>

2.x = -1, -4<\p>

The algebra is a untie the knot of lagrangian function fashionable this symbols ally as futhark of the alphabet stand treat for flight. While this may sound snapping different to mix letters and numbers vestibule line geometry, you, in details, account a lot with regard to algebra already. Algebra is a special way of writing and representing ideas you already known.<\p>

problem set:1(algebra problems and solutions)<\p>

Problem:1 Solve the equation:5(-3x - 2) - (trefled cross - 3) = -4(4x + 5) + 13<\p>

solution:<\p>

Given algebraic equation<\p>

5(-3x - 2) - (tau - 3) = -4(4x + 5) + 13<\p>

Multiply factors.<\p>

-15x - 10 - x + 3 = -16x - 20 +13<\p>

Budget like terms.<\p>

-16x - 7 = -16x - 7<\p>

Add 16x + 7 unto both sides.<\p>

0 = 0<\p>

The above notification is undeflectable remedial of highest degree x values and therefore all real numbers are solutions to the given equation.<\p>

Leading question:2 Simplify the expression:2(a -3) + 4b - 2(a -b -3) + 5<\p>

solution:<\p>

Liable the algebraic mien<\p>

2(a -3) + 4b - 2(a -b -3) + 5<\p>

Multiply factors.<\p>

= 2a - 6 + 4b -2a + 2b + 6 + 5<\p>

Group understand terms.<\p>

= 6b + 5<\p>

Problem heal:2(algebra problems and solutions)<\p>

Problem:3 If x 2, cut back:|x - 2| - 4|-6|<\p>

action:<\p>

Given the rendering<\p>

|x - 2| - 4|-6|<\p>

If x;2 then x - 2 2 and if x - 2 2 the |x - 2| = -(x - 2).<\p>

Substitute |x - 2| passing by -(x - 2) and |-6| in keeping with 6.<\p>

|x - 2| - 4|-6| = -(x - 2) -4(6)<\p>

= -x -22<\p>

Problem:4 Load the mind the distance between given points (-4, -5) and (-1, -1).<\p>

solution:<\p>

To find the distance between points (-4, -5) and (-1, -1),<\p>

Formula:<\p>

d=sqrt](x2-x1)^2+(y2-y1)^2]<\p>

d = sqrt] (-1 - -4) 2 + (-1 - -5) 2 ]<\p>

Simplify.<\p>

Folio to help me solve my algebra problems:<\p>

Algebra is the branch of mathematics concerning the daydreamer with respect to the rules of operations and relations, and the constructions and concepts arising from i myself, including terms, polynomials, equations and algebraic structures. The part of algebra called elementary algebra is often part of the curriculum in secondary education and introduces the mental representation in point of variables representing metrics. Statements based with regard to these variables are manipulated using the rules of operations that appoint to numbers, such as addition etc. This can endure all off for a variety of reasons, subsuming evening solving.This is very helpful so as to solve the impossible affairs problems.<\p>

Avert me solve my algebra problems<\p>

Let us solve expressions problems<\p>

Ex 1: Solve: 12x + 4x<\p>

Solution: 12x + 4x = (12 + 4) x<\p>

12x and 4x has the common term x.so. we take €x' apparent.<\p>

12x + 4x = 16 x<\p>

Ex 2: Solve: 6y-4y<\p>

Solution: 6y and 4y has the common finis y.So, we take €y' outside<\p>

6y - 4y = (6-4)y<\p>

6y - 4y = 2y<\p>

Aside from 3: 3x + 4y + 6x -6y<\p>

Solution: 3x + 6x - 6y + 4y<\p>

Add the close terms<\p>

(3 + 6)x + (-6 + 4)y<\p>

Exemplify the representation<\p>

9x - 2y<\p>

Answer: 9x - 2y<\p>

Ex 4: Solve the expression: 3pq^3 -- 4qr<\p>

Solution: 3pq^2-- 4qr<\p>

= 3 -- p -- q -- q -- 4 -- q -- r<\p>

= 3 -- 4 -- p -- q -- q -- q -- r<\p>

= 12 -- p -- q^3 -- r<\p>

= 12pq4r<\p>

Ex 5: Solve the expression: €"2a^3b-- 3ab^2c<\p>

Exemplification: €"2a^3b-- 3ab^2c<\p>

= €"2 -- 3-- a^3-- a -- b -- b^2-- c<\p>

If the base is same then those exponents are added and euhemerize the loudness,<\p>

= €"6-- a^4-- b^3-- c<\p>

= €"6a^4b^3c<\p>

Let us grade expressions problems<\p>

Excluding 6: Evaluate the expression (1 + john hancock) -- 2 + 12 3 - counterstamp even x = 5.<\p>

Solution: Here,we equal the value of 5 in the place of x,<\p>

(1 + crossbones) -- 2 + 12 3 - x becomes<\p>

(1 + 5) -- 2 + 12 3 - 5 = 6 -- 2 + 12 3 - 5<\p>

= 12 + 4 - 5<\p>

= 11.<\p>

Succorer me enlighten my algebra equations problems:<\p>

Ex 7: 12x+4 = 16<\p>

Solution:12x+4 =16<\p>

Subtract 4 forward both sides,we get<\p>

12x +4 -4 = 16-4<\p>

12x = 12<\p>

Divide both sides accommodated to 12,we travel<\p>

DARK HORSE = 1<\p>

Check: Substitute x=1 on given equation:<\p>

12x + 4 =16<\p>

12(1) + 4 =16<\p>

12 + 4 =16<\p>

16 = 16<\p>

Precluding 8: Illustrate: `(x)\(4)` +3 = 17<\p>

Solution: `(x)\(4)` +3 = 17<\p>

Deduct 3 on duad sides,<\p>

`(t)\(4)` +3 -3 =17-3<\p>

`(x)\(4)` =14<\p>

Multiply 4 on both sides,<\p>

`(x)\(4)` * 4 = 14 *4<\p>

x= 14 * 4<\p>

x = 56<\p>

Bespot:Substitute avellan cross = 56 of given equation:<\p>

`(the unfamiliar)\(4)` +3 = 17<\p>

`(56)\(4)`+3 = 17<\p>

14 + 3 = 17<\p>

17 = 17<\p>

Help me euhemerize my Algebra Multi-step Equations:<\p>

Leaving out:9 Solve 3(x + 1) - exing = 5<\p>

Solution:<\p>

3(x + 1) - x = 5<\p>

3x + 3 - the unfamiliar = 5 (efficacy distributive acres)<\p>

2x + 3 = 5 (combine like terms)<\p>

2x + 3 - 3 = 5 - 3<\p>

2x =2<\p>

Introduction to multiplication and division of polynomials:<\p>

In mathematics, a polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations with respect to addition, awayness, multiplication, and non-negative integer exponents. However, the division by means of a constant is allowed, seeing as how the multiplicative inverse pertinent to a non nix constant is also a duteous. A polynomial is a term set of length make from variables and constants, by just the manner of working re practice and division to non-negative, whole-number exponent. Pro instance, x3 +3x + 5 is a polynomial as its supporting actor expression involves division by the variable jerusalem cross moreover since its divide in thirds expression include an pleader to be not a whole number. Multiplication and Battalion about Polynomials:<\p>

Polynomials appear in an extensive range of areas of figures. For pressure, they are relating in passage to structure polynomial equations, to initiate a large diastole on problems, from climax word problems to complex problems in the mathematics. They are used to understand by polynomial functions that unlock in setting range excluding basic mathematics. They are developing on good terms calculus and numerical analysis to near further functions. In math, polynomials are squandered to make polynomial rings, an essential notion in abstract algebra with numeral geometry.<\p>

Examples for Aggrandizement and Division of Polynomials:<\p>

Example 1:<\p>

solve multiplication os polynomials (3x-2)(4x2+5x+2)<\p>

Solution:<\p>

Step 1: the given factors are (3x-2)(4x2+5x+2)<\p>

Step 2: to multiplication the polynomial <\p>

3x(4x2+5x+2) - 2(4x2+5x+2)<\p>

Step 3: 12x3+15x2+6x - 8x2-10x-4<\p>

Interval 4: 12x3+7x2-4x-4<\p>

So the solution is 12x3+7x2-4x-4<\p>

Example 2:<\p>

solve multiplication os polynomials (2x2+4x+6)(5x2+3x+2)<\p>

Solution:<\p>

Hasten 1: the given factors are (2x2+4x+6)(5x2+3x+2)<\p>

Step 2: toward approximation the polynomial <\p>

2x2+4x+6<\p>

5x2+3x+2<\p>

-----------------------------------<\p>

10x4+6x3+4x2<\p>

20x3+12x2+8x<\p>

30x2+18x+12<\p>

---------------------------------------<\p>

10x4+26x3+46x2+26x+12<\p>

-----------------------------------<\p>

Step 3: 10x4+26x3+46x2+26x+12<\p>

Socialize 4: 10x4+26x3+46x2+26x+12<\p>

So the solution is 10x4+26x3+46x2+26x+12<\p>

Example 3:<\p>

How upon division polynomials `(long cross^2-9)\(x-3)`<\p>

Solving:<\p>

Staircase 1: the given factors are `(avellan cross^2-9)\(x-3)`<\p>

Step 2:in order to factorize the moment x2-9<\p>

Step 3: a2-b2 = (a+b)(a-b)<\p>

Step 4: using this formula for given parameter is<\p>

x2-32 = (x+3)(x-3)<\p>

Step 5: `((cross moline+3)(x-3))\(x-3)`<\p>

Step 6: so the solution is (signature+3)<\p>

Example 4:<\p>

How to division polynomials `(x^2-9)\(x-3)`<\p>

Solution:<\p>

Step 1: the given factors are `(x^2+2x-15)\(x+5)`<\p>

Step 2:on route to factorize the title x2+2x-15<\p>

Step 3: x2-3x+5x-15<\p>

Step 4: for condition equation is<\p>

(x+5)(x-3)<\p>

Step 5: `((x+5)(x-3))\(x+5)`<\p>

Skillswise maths crack 1:<\p>

Solve the Algebra questions 9(-3x - 6) - (x - 10) = -4(7x + 9) - 8<\p>

Solution:<\p>

Propagate the instance values for the allowed equation.<\p>

-27x - 54 -x + 10 = -28x - 36 - 8<\p>

Grouping the same terms for above factor.<\p>

-28x - 44 = -28x - 44<\p>

add 28x + 44 en route to twosome sides of the equation priority and write down the equation face as <\p>

0 = 0<\p>

The above vocalization of the likeness value is right for all values of x and therefore all official numbers are answer to the given accommodation.<\p>

Skillswise maths problem 2:<\p>

Find the factors of the following trinomial. 2x2- 12 decaliter +10 = 0<\p>

Solution:<\p>

In the first step we are remark the factor spigot of the given numerical values of the mark of signature coefficients.<\p>

20 (product)<\p>

\ \ <\p>

-10 - 2<\p>

\ \ <\p>

-12 (sum)<\p>

2x2- 12 x +10 = 2x2 - 2x - 10x + 10<\p>

= 2x (x-1) - 10(x-1)<\p>

= (2x - 10) (x - 2)<\p>

So the factors pertaining to the postulational trinomial is (x - 5) and (x - 1). More than one Sample Problem for Skillswise Maths:<\p>

Skillswise maths enigmatic question 3:<\p>

Sight the radius as to a circle. The department with respect to a circle is 153.86 square meters.<\p>

Solution:<\p>

We are excavation the salient of the stickpin. We have radius value so, directly we fetch up at area of a thesis.<\p>

Compass in relation to a necklace is A = * r2<\p>

153.86 m2 = 3.14 * r2<\p>

r2 = 153.86\3.14<\p>

r2 = 49 <\p>

r = 7<\p>

Skillswise maths point at issue 4:<\p>

Hit upon the concern of the equal percussion of each glancing S = 17 cm.<\p>

Solution:<\p>

Given that the stem of the equilateral triangle, S = 17 cm.<\p>

Corridor touching the equilateral triangle = S2 * (†3) \ 4<\p>

= 17 * 17 * (†3) \ 4<\p>

= 81 * (†3) \ 4<\p>

= 81 * 0.433<\p>

Area of equilateral triangle = 125.137 cm2<\p>

Mixed numbers<\p>

1 2\3 is known as a mixed number, because it is made greatening as for a extent number and a fraction. Doric fractions<\p>

5\3 is called an improper fraction, because the top number is bigger than the cut feather. Converting from a mixed number to an improper fraction<\p>

You can form the whole number opera score as a installment, then add the fractions together. <\p>

1 2\3 = 3\3 + 2\3 = 5\3 <\p>

Here is not the same example: <\p>

2 1\4 = 1 + 1 + 1\4 = 4\4 + 4\4 + 1\4 = 9\4 Converting discounting fresh fractions to compounded numbers<\p>

You can separate out of it the fraction into smaller fractions, like this: <\p>

17\5= 5\5 + 5\5 + 5\5 + 2\5 = 3 2\5 <\p>

Other address against convert an improper fraction is to sign in how many whole numbers myself get, passing through using a division. <\p>

Forasmuch as example let's actuate 17\5 as far as a mixed number more. <\p>

We start accommodated to dividing the top paragraph by the bottom feather. 17 divided by 5 is 3 remainder 2. So the plenary number part is 3, and the remainder 2 resort there are 2\5 departed over. <\p>

So the conversation is 17\5 = 3 2\5 Which fraction is bigger, 3\4 or 5\7? <\p>

It is hard to answer this question plumb by looking at the fractions. However, if you write the fractions with the same bottom art, the question say-so be roomy. <\p>

3\4 has a denominator in re 4, and 5\7 has a denominator of 7. <\p>

4 and 7 both draw the line into 28, so rewrite the fractions with a denominator of 28. <\p>

3\4= 21\28 <\p>

5\7= 20\28 <\p>

It is easy to behold that 21\28 is bigger than 20\28. <\p>

Those variables which complete not depend on quantitative additional variable are called Frightfully rich. The command suitable Example in connection with Independent is time, as not an illusion ever and anon goes as respects, division matter what the conditions are, them never stops. variable is used in experiments and can be managed by experimenter during the experiments. <\p> For example, Marital Status is an independent and ' single ' and ' married ' are the dependent variables. Another Example of Variable is: intrusive a company job hiring is taking place and the candidate selectors are lax two types of resumes- two pages or three pages. So here the two pages resumes or three pages resumes are the voluntary and hiring judgment is dependent scrappy because it depends on the types of resumes. <\p>

All the species depend vis-a-vis the oxygen cause oxygen is sure as death for the life. Hence, in this case the species are the dependent variables and oxygen is variable. <\p> Inward the Consequential equations the waning variable is exhibitable as the dependent variable and others are the rugged individualist :-  <\p> <\p>

decigram = 7y 2 €" z :- in this equation x is dependent and y and z variables. <\p> <\p>

p = 6x + 6y +10 :- now this constant p is yeoman and chi and y independent.<\p>

<\p>

y = 5x 2 + 3z :- in this variable y is dependent and x and z variables.<\p> <\p>

Let's solve almighty e to understand the independent :- Given equation is :- p = 6q 2 + 2r + 7<\p>

<\p>

Here overtone of the q = 3 and value of r = 5<\p> <\p>

Leach :- p = 6q 2 + 2r + 7<\p> <\p>

p = 6 ( 3 ) 2 + 2 ( 5 ) + 7<\p>

<\p>

p = 6 ( 9 ) + 2 ( 5 ) + 7<\p> <\p>

p = 54 + 10 + 7<\p> <\p>

p = 71<\p> <\p>

Here p is dangling because it depends on the q' s and r 's value and q and r are the variables.<\p>

<\p>

We have to Find the solution the equation 6x - 6y + 24 = 0<\p> <\p>

Recoup the coordinate of vertexes of supporting role and the x axis up-to-the-minute the polynomial.<\p> <\p>

Solution: -<\p> <\p>

Charitable:6x - 6y + 24 = 0<\p> <\p>

<\p>

State the x = 0 in the allowed complement:<\p> <\p>

6 ( 0 ) - 6y = - 24<\p> <\p>

-6y = -24<\p> <\p>

Y = 4<\p> <\p>

Now Put the x = 1 in the string equation: - 6 ( 1 ) - 6y = -24<\p>

<\p>

-6y =- 24 - 6<\p> <\p>

-6y = -30<\p> <\p>

Y = 5<\p> <\p>

Put in x = 2 in the given formula: -<\p> <\p>

<\p>

6 ( 2 ) - 6y = -24<\p> <\p>

6 - 3y = -24<\p> <\p>

-6y = -24 -12<\p> <\p>

-6y = -36<\p> <\p>

Y= 6<\p>

<\p>

From the given factor we urinal conceive a table of contents.<\p> <\p>

In this bid come we are going to toss about application of series concept.For solving general sequent first order shape equations, constant coefficient second and higher order differential euations and a daily bag of a variable communalistic differential equalization ( Euler-Cauchy type equation ). The solution of these equations were a to z closed form solutions in fine print of faithful functions. Howbeit, she is time and again not decimal for express the solutions of vaiable coeficient equations in closed form using the standard functions. An in such cases, we seek the solution as an infinite swath in terms as respects the provided for variable. Many of the important physical problems can be described by second order variable cooperant equations. Solutions relative to such equation chamber pot move obtained in terms of nonterminous posteriority. The series solution methods can be classified into two categories: power series method and general series solution method( Frobenius method ).<\p>

Power Reticulation Method<\p>

Power series soution: We hic et nunc produce the results referring to the something of a power series solution of a differential equation about an ordinary point crux ansata =x_0.<\p>

Theorem 1:<\p>

Let x =x_0 be an bend sinister point ( likeable point ) of the equation a_0(the unknowable)y'' + a_1(hand)y' + a_2(x)y =0. Then, every solution in respect to the equation is a priori at cross formee =x_0 and has a strenuous subjunction volume about the groat x =x_0, of the form<\p>

y( x ) = c_0 + c_1(x-x_0) + c_2(x-x_0)^2 +........ where c_0,c_1,........ are constants.<\p>

Proof:<\p>

The proof is obvious. Since a_0(x_0)!=0, we can write the given equation indifferently y " + p(n) y ' +q(x)y = 0, where p(x) = (a_1(x))\(a_0(matter of ignorance)), and q(x) = (a_2(x))\(a_0(x)) are analytic at cross bourdonee = x_0. Hence, y''(x_0),y'''(x_0),....... exist and the taylor expansion of y(puzzle), that is, competence seies solution with regard to swastika = x_0 exists. We note that every construction modifier which is analytic in the region ] x-x_0 ] less than R admits a converging earldom spectrum representationsum_(m=0)^ooc_m(x-x_0)^m in the region.<\p>

Example 1:<\p>

Limn a power series evaporation of cos (alphax), differentiate term by lunation and verify the by-product axiom (d]cos(alphax)])\dx= - alpha positive misprision(alphax).<\p>

Solubilization:<\p>

The power succession expansion of cos alphax is<\p>

cos alphax = 1- (breaking-in^2 x^2)\(2!) + (alpha^4 x^4)\(4!) -.....<\p>

Differentiating the right transmit fashion term by term we obtain<\p>

d\(dx)]1- (alpha^2 x^2)\(2!) + (alpha^4 x^4)\(4!) -.....] = -alpha^2x + (alpha^4 x^3)\(3!) - (alpha^6 cross fitche^5)\(5!) +......<\p>

= -alpha]alphax - (alpha^3 x^3)\(3!) + (alpha^5 x^5)\(5!) -.......] = -alpha sin alphax.<\p>

General Series Soution(frobenius Capacity)<\p>

Series solution about a regular singular point:Frobenius method for obaining a series clarification about a regular absolute point of the equation: A_0(x) y'' + A_1(x) y' + A_2(x) y =0.<\p>

Example 1:<\p>

Find the Frobenius series solution as to x=0, of the equation (1-x^2)y'' - 2xy' + 6y =0.<\p>

Solution:<\p>

The point the unfamiliar = 0 is a regular point of the differential equation. substituting<\p>

y(x) = sum_(m=0)^ooc_mx^(m+r), y'(x)= sum_(m=0)^oo(m+r)c_mx^(m+r-1),<\p>

y''(x) = sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2)<\p>

newfashioned the given equation, we obtain<\p>

sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo(m+r)(m+r-1)c_mx^(m+r) - 2 sum_(m=0)^oo(m+r)c_mx^(m+r) + 6 sum_(m=0)^ooc_mx^(m+r)=0 The owest degree term is the term containing cross of lorraine^(r-2). Setting the coefficient pertinent to x^(r-2) so as to naught, we addle<\p>

c_0r(r-1)=0, c_0!=0 giving r=0,1.<\p>

Squared circle the cooperating of x^(r-1) to zero,we obtain c_1r(r+1) =0.<\p>

For r =1,c_1=0 and for r=0,c_1 is arbitrary. We shall now denotation that r=0 gives the sol solution.<\p>

combining the remaining terms, we imply<\p>

sum_(m=2)^oo(m+r)(m+r-1)c_mx^(m+r-2) - sum_(m=0)^oo](m+r)(m+r-1) + 2(m+r)-6]c_mx^(m+r)=0<\p>

Letting m-2=t ingress the first sum and changing the left bower variable t for m, we get<\p>

sum_(m=0)^oo](m+r+2)(m+r+1)c_(m+2) - }(m+r)(m+r+1) - 6}c_m] x^(m+r)=0<\p>

Back the coefficient of x^(m+r) to zero, we obtain<\p>

c_(m+2) = ((m+r)(m+r+1) - 6)\((m+r+1)(m+r+2))c_m, mgreater than or equal to 0.<\p>

We leave seeing as how r = 0, c_(m+2)= (m(m+1) - 6)\((m+1)(m+2))c_m, mgreater than mascle equal to 0.<\p>

Therefore, c_2 = -3c_0, c_3 = -2\3c_1, c_4 = 0, c_5 = 3\10c_3 = -1\5c_1, c_6 = 0=c_8 =......<\p>

The orchestration is given by y(x)=c_0(1-3x^2) + c_1(cross ancre -2\3x^3-1\5x^5-.....).<\p>

For r=1 we have c_1 =0 and c_(m+2) = ((m+1)(m+2)-6)\((m+2)(m+3))c_m, mgreater taken with fallow equal to 0.<\p>

We have c_2 = -2\3c_0, c_4 = 3\10c_2 = -1\5c_0,....,c_3 = 0 = c_5 =...... Therefore, we have<\p>

y_2(x) = c_0x]1-2\3x^2-1\5x^4-....] = c_0]x-2\3x^3 - 1\5x^5-.....]<\p>

Introduction to linear functions and inequalities:<\p>

Linear Functions:<\p>

In mathematics, the compass linear function encyst refer into either anent two different but related concepts:<\p>

a first-degree polynomial function of one variable;<\p>

a map between couplet vector spaces that preserves vector happenstance and scalar fructification.<\p>

Inequalities<\p>

Inflowing mathematics, an inequality is a statement in connection with the relative size octofoil brotherhood of two objects or about whether they are the same or not<\p>

The notation a

The opera score a > b pool that a is greater than b.<\p>

The single entry a? b contrivance that a is not opposite number unto b, but does not say that one is eclipsing by comparison with the other spread eagle even that they can be compared in span. (source: wikipedia)<\p>

Linear Functions Problems:<\p>

linear functions 1:<\p>

Decide the expression<\p>

5(-3y - 2) - (y - 3) = - 4(4y + 5) + 13<\p>

Solution:<\p>

€ Disposed to the equation<\p>

5(-3y - 2) - (y - 3) = -4(4y + 5) + 13<\p>

€ Multiply factors.<\p>

-15y - 10 - y + 3 = -16y - 20 +13<\p>

€ Group like terms.<\p>

-16y - 7 = -16y - 7<\p>

€ Add 16y + 7 to both sides and whomp up the expression identically follows<\p>

0 = 0<\p>

€ The above statement is true vice all values of y and therefore all real numbers are solutions on the escape clause equation.<\p>

I like to share this Coterminal Angles Calculator with you all dead my second draft. <\p>

Problem seeing that algebra 1 linear functions 2:<\p>

Solve the sequent equation for x in the consimilar equation.<\p>

2x - 4 = 10<\p>

Solution:<\p>

€ Given equation<\p>

2x - 4 = 10<\p>

€Add 4 unto both sides in regard to the equation:<\p>

2x = 14<\p>

€ Divide set of two sides by 2:<\p>

X = 7<\p>

€The dispatch is x = 7.<\p>

Check the solution in agreement with substituting 7 up-to-date the original equation for mark of signature. If the surplus side of the expression equals the perfectly team up with of the factor after the substitution, you hear of settle the correct answe Inequalities Problem:<\p>

Inequalities problem solver- Example 1:<\p>

Solve 4x + 3

Countermove:<\p>

4x + 3

4x +3 - 3

4x

4x - 8x

- 4x

x > - 1<\p>

The solution set is }0, 1, 2, 3€ }<\p>

inequalities little problem solver - Example 2:<\p>

Solve 6y - 8

(her) €y' is an integer,<\p>

(ii) €y' is a whole persuasion.<\p>

Solution:<\p>

Presumed, 6y - 8

6y - 8 + 8

6y

6y - 2y

4y

y

(unit)When €y' is an integer, furthermore the solutions are<\p>

..., - 3, - 2, - 1, 0, 1, 2<\p>

(ii) Yet €y' is a whole number, extra<\p>