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Find all real solutions of (x2 – 43)/(x2 – 4x – 5) = 5/(x + 1) – 3/(x-5)

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  • Last Updated : 16 Dec, 2021

The concept of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are termed here as variables. this expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is termed a coefficient.

An idea of expressing numbers using letters or alphabets without specifying their actual values is defined as an algebraic expression.

What is an Algebraic Expression?

It is an expression that is made up of variables and constants along with algebraic operations such as addition, subtraction, etc.. these Expressions are made up of terms. Algebraic expressions are the equations when the operations such as addition, subtraction, multiplication, division, etc. are operated upon any variable.

A combination of terms by the operations such as addition, subtraction, multiplication, division, etc is termed as An algebraic expression (or) a variable expression.

Examples: 2x + 4y – 7, 3x – 10, etc.

The above expressions are represented with the help of unknown variables, constants, and coefficients. The combination of these three terms is termed as an expression. unlike the algebraic equation, It has no sides or ‘equals to’ sign.

Types of Algebraic Expressions

Based on the number of terms in the expression, algebraic expressions are divided into 3 parts:

  • Monomial Expression
  • Binomial Expression
  • Polynomial Expression

Monomial Expression

An expression that has only one term is termed a Monomial expression.

Examples of monomial expressions include 5x4, 3xy, 2x, 5y, etc.

Binomial Expression

An algebraic expression which is having two terms and unlike are termed as a binomial expression

Examples of binomial include 2xy + 8, xyz + x2, etc.

Polynomial Expression

An expression that has more than one term with non-negative integral exponents of a variable is termed a polynomial expression.      

Examples of polynomial expression include ax + by + ca, x3 + 5x + 3, etc.

Some Other Types of Expression

We have other expressions also apart from monomial, binomial, and polynomial types of expressions which are

  • Numeric Expression
  • Variable Expression

Numeric Expression

An expression that consists of only numbers and operations, but never includes any variable is termed a numeric expression.

Some of the examples of numeric expressions are 14 + 5, 18 ÷ 2, etc.

Variable Expression

An expression that contains variables along with numbers and operations to define an expression is termed as A variable expression.

Some examples of a variable expression include 4x + y, 5ab + 53, etc.

Some Important Algebraic Formulas

There are some terms of algebraic expression which basically used,

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

(a + b)(a – b) = a2 – b2

(x + a)(x + b) = x2 + x(a + b) + ab

(a + b)3 = a3 + b3 + 3ab(a + b)

(a – b)3 = a3 – b3 – 3ab(a – b)

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)

Example: If 2x2+3xy+4x+7 is an algebraic expression. Determine the equation.

Solution:

2x2, 3xy, 4x, and 7 are the Terms

Coefficient of term: 2 is the coefficient of x2

Constant term: 7

Variables: here x, y are variables

Factors of a term: If 2xy is a term, then its factors are 2, x, and y.

Like and Unlike terms : Example of like and unlike terms:

  • Like terms: 4x and 3x
  • Unlike terms: 2x and 4y

Find all real solutions of \frac{x^2 - 43}{x^2 - 4x - 5} = \frac{5}{x + 1} - \frac{3}{x-5}

Solution:

Given that:

\frac{x^2 - 43}{x^2 - 4x - 5} = \frac{5}{x + 1} - \frac{3}{x-5}

By simplifying and factorizing above terms 

\frac{x^2 - 43}{x^2 - 4x - 5} = \frac{5}{x + 1} - \frac{3}{x-5}

\frac{x^2 - 43}{(x+1)(x-5)} = \frac{5(x-5)-3(x+1)}{(x+1)(x-5)}

Multiplying both sides by (x+1)(x-5) then we will get

{x2 – 43} = {5(x-5) – 3(x+1)}

x2 – 43 = 5x – 25 – (3x + 3)

x2 – 43 = 5x – 25 – 3x – 3

x2 – 43 = 2x – 28

x2 – 43 – 2x + 28 = 0

x2 – 2x – 15 = 0

By factorizing the term

(x+3) (x – 5) = 0

therefore 

x + 3 = 0 

x = -3 or x – 5 = 0 

x = 5 

So the real solution for the above equations are -3, 5

Similar Questions 

Question 1: Factorize 4a2 – 9b2 -2a -3b?

Solution:

Given : 4a2 – 9b2 – 2a – 3b

 = {(2a)2 – (3b)2 – (2a + 3b)}

 = (2a – 3b) (2a + 3b) – (2a + 3b)

 = (2a + 3b) {(2a – 3b) – 1}

 = (2a + 3b) (2a – 3b – 1)

Question 2: Factorize x4 + x2 + 1

Solution:

Given: x4 + x2 +1

Add and subtract x2 in above term

= x4 + x2 + 1 + x2 – x2

= x4 + x2 + x2 + 1 – x2

= (x4 + 2x2 + 1) – x2

= (x2 + 1)2 – x2           

= (x2 + 1 + x )(x2 + 1 – x)

= (x2 + x + 1) (x2 – x + 1)

Question 3: Solve for y in the equation: \frac{-1}{y} = -0.25x^2 - 5.5

Solution:

Given: {-1}/{y} = -0.25x2 – 5.5

Multiply both sides by y

⇒ (-1/y)(y) = y(-0.25x2 – 5.5)

⇒ -1 = -0.25x2y – 5.5y    

⇒ -1 = y(-0.25x2 – 5.5)

⇒ -1 = -y(-0.25x2 – 5.5)

⇒ y = 1/(-0.25x2 – 5.5)

Question 4: Simplify 49x2 – 25y2

Solution:

Given, 49x2 – 25y2

= 49x2 – 25y2

= (7x)2 – (5y)2

Now, x2 – y2 =  (x + y)(x – y)

= (7x + 5y)(7x – 5y)         


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