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Basic Algebraic Identities

Basic Algebraic Identities

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Algebraic Identities of Square


 »  Understanding Identities
All algebraic identities use the following.
`color(coral)((p+q))xx color(deepskyblue)((b+c))`
`= color(coral)(p) color(deepskyblue)(b) + color(coral)(q) color(deepskyblue)(b) ``+ color(coral)(p) color(deepskyblue)(c) + color(coral)(q) color(deepskyblue)(c)`


 »  Algebraic Identities of Square


   `(x+y)^2 ``= x^2+2xy+y^2`


   `(x-y)^2 ``= x^2-2xy+y^2`


   `(x+y)(x-y) ``= x^2-y^2`


   `(x+a)(x+b) ``= x^2+(a+b)x+ab`


   `(x+y+z)^2 ``= x^2+y^2+z^2``+2xy+2yz+2zx`

Algebraic Identities of Square

plain and simple summary

nub

plain and simple summary

nub

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algebraic identities are derived by expanding the multiplication in algebraic expressions.

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simple steps to build the foundation

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In this page, standard algebraic identities with exponent 2 are introduced and explained.


Keep tapping on the content to continue learning.
Starting on learning "Algebraic Identities of Square". ;; In this page, standard algebraic identities with exponent 2 are introduced and explained.

What does 'identity' mean?

  • It is just a name.
  • equality of two expressions; left and right hand side are identical

Answer is 'equality of two expressions; left and right hand side are identical'

Identities are statements that specify two expressions as equal.

In numerical terms, some of the identities are
 •  `2+3 = 5`
(left hand side `2+3` equals to right hand side `5`)
 •  `4-3 = 7/7`
(left hand side and right hand side are equal.)

Similar to identities of numerical expressions, algebraic identities are statements that specify two algebraic expressions as equal.
(left hand side equals right hand side)
 •  `a^2+2a^2+b = 3a^2+b`
 •  `(2+x)x = x^2+2x`

Some identities are repeatedly used. Such identities are called standard algebraic identities. While handling algebraic expressions, it helps to identify the left hand side or right hand sides of the standard identities.

Students are taught with these standard algebraic identities.

Which of the following equals `2xx(4+1)`?

  • `2xx4 + 1`
  • `2xx4 + 2 xx1`

The answer is '`2xx4 + 2 xx1`'.

This is verified by finding the value of the numerical expressions

Question: `2xx(4+1)``=2 xx5``=10`

first choice: `2xx4 + 1` `=8+1``=9`

second choice: `2xx4+2xx1` `=8+2``=10`

The expression given in question equals the second choice.

Generalizing the numbers to variables.
`axx(b+c) = axxb+axxc`.

Note that `a` is a measure of quantity given by a variable.
The variable `a` can be `4` or `2+2` or `3+1` or `(p+q)`.
This is given as
`color(coral)(a)xx color(deepskyblue)((b+c)) ``= color(coral)(a)xx color(deepskyblue)b+color(coral)(a)xx color(deepskyblue)c`

`color(coral)(4)xx color(deepskyblue)((b+c)) ``= color(coral)(4)xx color(deepskyblue)b+color(coral)(4)xx color(deepskyblue)c`

`color(coral)((2+2))xx color(deepskyblue)((b+c)) ``= color(coral)((2+2))xx color(deepskyblue)b+color(coral)((2+2))xx color(deepskyblue)c`

`color(coral)((3+1))xx color(deepskyblue)((b+c)) ``= color(coral)((3+1))xx color(deepskyblue)b+color(coral)((3+1))xx color(deepskyblue)c`

`color(coral)((p+q))xx color(deepskyblue)((b+c)) ``= color(coral)((p+q))xx color(deepskyblue)b + color(coral)((p+q)))xx color(deepskyblue)c`

We have understood that
`color(coral)((p+q)) xx color(deepskyblue)((b+c)) ``= color(coral)((p+q))xxcolor(deepskyblue)(b) + color(coral)((p+q))xxcolor(deepskyblue)(c)`

The same proof can be used to take in `color(deepskyblue)b` and `color(deepskyblue)(c)` into `color(coral)((p+q))`. This is given below.

`color(coral)((p+q))xx color(deepskyblue)((b+c))`
`= color(coral)((p+q))xxcolor(deepskyblue)(b) + color(coral)((p+q))xxcolor(deepskyblue)(c)`
`= color(coral)(p)xx color(deepskyblue)(b) + color(coral)(q)xxcolor(deepskyblue)(b) + color(coral)(p)xx color(deepskyblue)(c) + color(coral)(q)xxcolor(deepskyblue)(c)`

This is a very important result in understanding identities.
It is important for students to understand how this result is derived.

`color(coral)((p+q))xx color(deepskyblue)((b+c))`
`= color(coral)(p) color(deepskyblue)(b) + color(coral)(q)color(deepskyblue)(b) + color(coral)(p) color(deepskyblue)(c) + color(coral)(q)color(deepskyblue)(c)`

All algebraic identities use this result to prove that left-hand-side equals right-hand-side.

Do not memorize any identities by rote. Derive them quickly on the fly using this result.
With repeated use and practice, one will remember.

Which of the following is equivalent to `(a+b)^2`?

  • `(a+b)(a+b)`
  • `a+b^2`
  • `a^2+b^2`

The answer is '`(a+b)(a+b)`'

`(a+b)^2`

`= color(coral)((a+b))color(deepskyblue)((a+b))`

`= color(coral)((a+b))color(deepskyblue)(a) ``+ color(coral)((a+b))color(deepskyblue)(b)`

`= color(coral)(a)color(deepskyblue)(a) + color(coral)(b)color(deepskyblue)(a) ``+ color(coral)(a)color(deepskyblue)(b)``+ color(coral)(b)color(deepskyblue)(b)`

`= a^2+ 2ab + b^2`

Which of the following is equivalent to `(a-b)^2`?

  • `(a-b)(a-b)`
  • `a-b^2`
  • `a^2-b^2`

The answer is '`(a-b)(a-b)`'

`(a-b)^2`

`= color(coral)((a-b))color(deepskyblue)((a-b))`

`= color(coral)((a-b))color(deepskyblue)(a) ``- color(coral)((a-b))color(deepskyblue)(b)`

`= color(coral)(a)color(deepskyblue)(a) - color(coral)(b)color(deepskyblue)(a) ``- color(coral)(a)color(deepskyblue)(b)``-(- color(coral)(b)color(deepskyblue)(b))`

`= a^2 - 2ab + b^2`

Comparing the two identities

`(a_1+b_2)^2 = a_1^2+2a_1b_1+b_1^2`
`(a_2-b_2)^2 = a_2^2-2a_2b_2+b_2^2`

Substituting in the first identity `b_1 =- b_2`; the second identity is derived. Students may work this out to understand.

Which of the following is equivalent to `(a+b)(a-b)`?

  • `a(a-b)+b(a-b)`
  • `aa-ba+ba-b b`
  • `a^2-b^2`
  • all the above

The answer is 'all the above'

`color(coral)((a+b))color(deepskyblue)((a-b))`

`= color(coral)((a+b))color(deepskyblue)(a) ``- color(coral)((a+b))color(deepskyblue)(b)`

`= color(coral)(a)color(deepskyblue)(a) + color(coral)(b)color(deepskyblue)(a) ``- color(coral)(a)color(deepskyblue)(b)``- color(coral)(b)color(deepskyblue)(b)`

`= a^2 - b^2`

Which of the following is equivalent to `(x+a)(x+b)`?

  • `x+ax+b`
  • `x+ab`
  • `x^2+(a+b)x+ab`
  • `x+ax+ab`

The answer is '`x^2+(a+b)x+ab`'

`color(coral)((x+a))color(deepskyblue)((x+b))`

`= color(coral)((x+a))color(deepskyblue)(x) ``+ color(coral)((x+a))color(deepskyblue)(b)`

`= color(coral)(x)color(deepskyblue)(x) + color(coral)(a)color(deepskyblue)(x) ``+ color(coral)(x)color(deepskyblue)(b)``+ color(coral)(a)color(deepskyblue)(b)`

`= x^2+ (a+b)x + ab`

Comparing the two identities

`(x+a_1)(x+b_1) ``= x^2+ (a_1+b_1)x + a_1b_1`

`(a_2+b_2)^2 ``= a_2^2+2a_2b_2+b_2^2`


Substituting in the first identity `x=a_2` ; `a_1=b_2`; and `b_1 = b_2`; the second identity is derived. Students may work this out to understand.

It is known that `(a+b)^2 = a^2+2ab+b^2`, can this result be used to find `(a+b+c)^2`?

  • Yes. Consider `a+b+c = a+ (b+c)` where `b+c` is a single number
  • No. It is a new identity

The answer is 'Yes. Consider `a+b+c = a+ (b+c)` where `b+c` is a single number'

`(color(coral)(a+)color(deepskyblue)((b+c))) ^2`

`= color(coral)(a^2)+ 2color(coral)(a)color(deepskyblue)((b+c)) ``+ color(deepskyblue)((b+c)^2)`

`= color(coral)(a^2) + 2color(coral)(a)color(deepskyblue)(b) ``+ 2color(coral)(a)color(deepskyblue)(c)``+ color(deepskyblue)(b^2+2bc+c^2)`

`= a^2+b^2+c^2+2ab+2bc+2ca`

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Algebraic Identities of Square :

`(x+y)^2 ``= x^2+2xy+y^2`

`(x-y)^2 ``= x^2-2xy+y^2`

`(x+y)(x-y) ``= x^2-y^2`

`(x+a)(x+b) ``= x^2+(a+b)x+ab`

`(x+y+z)^2 ``= x^2+y^2+z^2``+2xy+2yz+2zx`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

What is `(a+b-c)^2`?

  • `a^2+2a(b-c)+(b-c)^2`
  • `a^2+b^2+c^2+2ab``+2bxx(-c)+2xx(-c)xxa`
  • `a^2+b^2+c^2+2ab-2bc-2ca`
  • all the above

The answer is 'all the above'

your progress details

Progress

About you

Progress

What does 'identity' mean?
just;name
It is just a name.
equality;2;expressions;left;right
equality of two expressions; left and right hand side are identical
Answer is 'equality of two expressions; left and right hand side are identical'
Identities are statements that specify two expressions as equal. ;; In numerical terms, some of the identities are, 2 + 3 = 5 ; left hand side 2+3 equals to right hand side 5. ;; 4 minus 3 equals 7 divided by 7. left hand side and right hand side are equal. Similar to identities of numerical expressions, algebraic identities are statements that specify two algebraic expressions as equal. Note: left hand side equals right hand side ;; a squared + 2 a squared + b, = 3 a squared + b ;; 2 + x multiplied x, = x squared + 2x.
Some identities are repeatedly used. Such identities are called standard algebraic identities. While handling algebraic expressions, it helps to identify the left hand side or right hand sides of the standard identities. Students are taught with these standard algebraic identities.
which of the following equals 2 multiplied, 4 + 1?
1
2
The answer is "2 times 4 plus 2 times 1"
Generalizing the numbers to variables. a multiplied b + c, = a multiplied b, + a multiplied c. ;; Note that a is a measure of quantity given by a variable. The variable a can be 4, or 2+2, or 3+1 or p + q. This is given in detail to understand.
We have understood that p+q multiplied b + c, equals p+q multiplied b, +, p+q multiplied c. ;; The same proof can be used to take in b and c into p plus q. This is given in detail. ;; This is a very important result in understanding identities.It is important for students to understand how this result is derived.
The result for p+q multiplied b + c is given as p b + q b + p c + q c. ;; All algebraic identities use this result to prove that left hand side equals right hand side. ;; Do not memorize th
Which of the following is equivalent to a + b whole squared?
1
2
3
The answer is "a + b multiplied a + b"
a + b whole squared identity is derived. It equals a squared + 2a b + b squared.
Which of the following is equivalent to a minus b whole squared?
1
2
3
The answer is "a - b multiplied a - b"
a minus b whole squared identity is derived. It equals a squared minus 2a b + b squared.
Comparing the two identities the second can be derived from the first.
Which of the following is equivalent to a plus b multiplied a minus b?
1
2
3
4
The answer is 'all the above'
a + b multiplied a - b identity is derived. It equals a squared minus b squared.
Which of the following is equivalent to x+a, multiplied, x+b ?
1
2
3
4
The answer is "x squared plus a+b x + a b"
x+a multiplied x+b identity is derived. It equals x squared +, (a+b)x + a b.
Comparing the two identities the second can be derived from the first.
It is known that a+b whole squared equals a squared + 2 a b + b squared. Can this result be used to fine a+b+c whole squared?
1
2
The answer is "Yes. Consider a + b + c equals a +, b+ c where b+c is a single number"
a + b + c whole squared identity is derived. It equals a squared + b squared + c squared + 2 a b + 2 b c + 2 c a.
algebraic identities are derived by expanding the multiplication in algebraic expressions.
Algebraic Identities of square:
what is a+b minus c whole squared?
1
2
3
4
The answer is 'all the above'

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