Identities
Identities are ways of expressing algebraic values in different ways. The values of the written expressions are identical. This is why the word ‘identical’ is used.
"Same thing Different form!"
Such following expressions and equations are identical:
\( \displaystyle 3x=2x+x \)
\( \displaystyle 5.(x+1)=5x+5 \)
\( \displaystyle (a+b)²=a²+2ab+b² \)
Identities are valid / true for all real numbers.
Let us choose a number for this expression \( \displaystyle 5.(x+1)=5x+5 \) and try.
Let \( \displaystyle x=8 \)
\( \displaystyle 5.(8+1)=5.8+5 \)
\( \displaystyle 5.(9)=40+5 \)
\( \displaystyle 45=45 \)
The left side is 45, the right side is 45, and they are equal to one another. Whatever number you give \( \displaystyle x \) the right and left side will equal to one another because the expressions are identical. They are the same.
"Same thing Different form!"
\( \displaystyle 5.(x+1)=5x+5 \)
\( \displaystyle (a+b)²=a²+2ab+b² \)
Identities are valid / true for all real numbers.
Let us choose a number for this expression \( \displaystyle 5.(x+1)=5x+5 \) and try.
Let \( \displaystyle x=8 \)
\( \displaystyle 5.(8+1)=5.8+5 \)
\( \displaystyle 5.(9)=40+5 \)
\( \displaystyle 45=45 \)
The left side is 45, the right side is 45, and they are equal to one another. Whatever number you give \( \displaystyle x \) the right and left side will equal to one another because the expressions are identical. They are the same.
Why do we need identities?
They can help us whilst calculating some mathematical equations.
Sometimes you can change an unsolvable equation into another form to calculate it easily.
Don’t think about this in purely mathematical terms, you can think of these as different compounds that show the same effect in chemistry or as ‘equivalent drugs’ – where two different medicines produce the same result or it is as if telling a foreigner the same thing you want to tell them in a different language in which they can understand.
They can help us whilst calculating some mathematical equations.
Sometimes you can change an unsolvable equation into another form to calculate it easily.
Don’t think about this in purely mathematical terms, you can think of these as different compounds that show the same effect in chemistry or as ‘equivalent drugs’ – where two different medicines produce the same result or it is as if telling a foreigner the same thing you want to tell them in a different language in which they can understand.
Don’t think about this in purely mathematical terms, you can think of these as different compounds that show the same effect in chemistry or as ‘equivalent drugs’ – where two different medicines produce the same result or it is as if telling a foreigner the same thing you want to tell them in a different language in which they can understand.
Identities or Equations?
Identities are valid for all real numbers, equations are valid for certain groups of real numbers or a certain amount of numbers. By defining equations you may find a result where the value is not clear. Just as \( \displaystyle x=5 \) , but with identities there is no such expression such as (\( \displaystyle x=5 \))
The goal of identities, is to express equations in different ways.
\( \displaystyle 3x+4=25 \) is an equation ,
\( \displaystyle x=7 \) is equal to
\( \displaystyle 3x+2x=5x \) It is identical;
all numerical values of \( \displaystyle x \)are real numbers.
Some important identities;
Identities are valid for all real numbers, equations are valid for certain groups of real numbers or a certain amount of numbers. By defining equations you may find a result where the value is not clear. Just as \( \displaystyle x=5 \) , but with identities there is no such expression such as (\( \displaystyle x=5 \))
The goal of identities, is to express equations in different ways.
\( \displaystyle 3x+4=25 \) is an equation ,
\( \displaystyle x=7 \) is equal to
\( \displaystyle 3x+2x=5x \) It is identical;
all numerical values of \( \displaystyle x \)are real numbers.
Some important identities;
Perfect square identities
The identity of \( \displaystyle (a+b)² \)
The square of anything is the multiplication of any number by itself. Let’s write it down within brackets next to one another.
Let’s simplify ; \( \displaystyle a²+2ab+b² \)
So ;
\( \displaystyle (a+b).(a+b)=a²+2ab+b² \)
\( \displaystyle (a+b)²=a²+2ab+b² \)
The identity of \( \displaystyle (a-b)² \)
\( \displaystyle (a-b) \) Let’s write it down within brackets next to one another.
Let’s simplify; \( \displaystyle a²-2ab+b² \)
so;
\( \displaystyle (a-b)²=a²-2ab+b² \)
Please go with link for explanation " Geometrically Modelling identities with area "
Memorizing these identities will help you answer questions quickly. If you have problems with memorizing don’t worry, just remember how to square any expression, write them down next to each another and then, multiply.
How to Memorize ?
\( \displaystyle (a+b)² \)
The square of the first, twice as much as the first and second multiplied , the square of the second
\( \displaystyle (a-b)² \)
The square of the first, twice as much as the first and second multiplied , the square of the second
Yes it is same 🙂
What Do We Mean by the First term and the Second?
With algebraic expressions such as \( \displaystyle (a+b) \), terms are taken together with their notations.We call \( \displaystyle a \) the first term (because the notation hasn’t been written in front of it +) \( \displaystyle +a \) is the first term, and \( \displaystyle +b \) is the second term.
\( \displaystyle (a+b) \) \( \displaystyle +a \) is first , \( \displaystyle +b \) is the second term
\( \displaystyle (a+b) \) \( \displaystyle +a \) is the first , \( \displaystyle -b \) is the second term
While calculating, the signs are not written directly!!
At the end of the calculation, the component is written together within the signs and all components are brought together.
You can obtain more information from our introduction to algebra page.
\( \displaystyle (a+b)² \)
The square of the first, the square of \( \displaystyle +a \), makes \( \displaystyle +a² \)
Twice as much as the value of the result of the multiplication of the first and second term, the first \( \displaystyle +a \) , second \( \displaystyle +b \), \( \displaystyle 2.(+a).(+b)=+2ab \)
The square of the second , the square of \( \displaystyle +b \) makes \( \displaystyle +b² \)
Lets combine them ;
\( \displaystyle (a+b)²=a²+2ab+b² \)
\( \displaystyle (a-b)² \)
The square of the first, the square of \( \displaystyle +a \) , makes \( \displaystyle +a² \)
Twice as much as the value of the result of the multiplication of the first and second term, the first \( \displaystyle +a \) , second \( \displaystyle -b \), \( \displaystyle 2.(+a).(-b)=-2ab \)
The square of the second , the square of \( \displaystyle -b \) makes \( \displaystyle +b² \)
Lets combine them ;
\( \displaystyle (a-b)²=a²-2ab+b² \)
Let’s write down a few examples;
\( \displaystyle (x+5)² \)
\( \displaystyle +x² \) , the square of the first term
\( \displaystyle +10x \) , Twice the multiplication of the first and second term
\( \displaystyle +25 \) , the square of the second term
\( \displaystyle x²+10x+25 \)
\( \displaystyle (x-5)² \)
\( \displaystyle +x² \) , the square of the first term
\( \displaystyle -10x \) , Twice the multiplication of the first and second term
\( \displaystyle +25 \) , the square of the second term
\( \displaystyle (3x-4)² \)
\( \displaystyle +9x² \) , the square of the first
\( \displaystyle -24x \) , Twice the multiplication of the first and second term
\( \displaystyle +16 \) , the square of the second
\( \displaystyle (3x-4)²=9x²-24x+16 \)
\( \displaystyle (-x+3)² \)
\( \displaystyle +x² \) , the square of the first
\( \displaystyle -6x \) , Twice the multiplication of the first and second term
\( \displaystyle +9 \) , the square of the second
\( \displaystyle (-x+3)²=x²-6x+9 \)
If you memorize by placing the notations first, you will make mistakes in these types of identities. You need to know how to separate the terms, notations will come after multiplication of the terms.
is a wrong way of writing it down !
The identity of \( \displaystyle (a+b)² \)
The square of anything is the multiplication of any number by itself. Let’s write it down within brackets next to one another.
Let’s simplify ; \( \displaystyle a²+2ab+b² \)
So ;
\( \displaystyle (a+b).(a+b)=a²+2ab+b² \)
\( \displaystyle (a+b)²=a²+2ab+b² \)
The identity of \( \displaystyle (a-b)² \)
\( \displaystyle (a-b) \) Let’s write it down within brackets next to one another.
Let’s simplify; \( \displaystyle a²-2ab+b² \)
so;
\( \displaystyle (a-b)²=a²-2ab+b² \)
Please go with link for explanation " Geometrically Modelling identities with area "
Memorizing these identities will help you answer questions quickly. If you have problems with memorizing don’t worry, just remember how to square any expression, write them down next to each another and then, multiply.
How to Memorize ?
\( \displaystyle (a+b)² \)
The square of the first, twice as much as the first and second multiplied , the square of the second
\( \displaystyle (a-b)² \)
The square of the first, twice as much as the first and second multiplied , the square of the second
Yes it is same 🙂
What Do We Mean by the First term and the Second?
With algebraic expressions such as \( \displaystyle (a+b) \), terms are taken together with their notations.We call \( \displaystyle a \) the first term (because the notation hasn’t been written in front of it +) \( \displaystyle +a \) is the first term, and \( \displaystyle +b \) is the second term.
\( \displaystyle (a+b) \) \( \displaystyle +a \) is first , \( \displaystyle +b \) is the second term
\( \displaystyle (a+b) \) \( \displaystyle +a \) is the first , \( \displaystyle -b \) is the second term
While calculating, the signs are not written directly!!
At the end of the calculation, the component is written together within the signs and all components are brought together.
You can obtain more information from our introduction to algebra page.
\( \displaystyle (a+b)² \)
The square of the first, the square of \( \displaystyle +a \), makes \( \displaystyle +a² \)
Twice as much as the value of the result of the multiplication of the first and second term, the first \( \displaystyle +a \) , second \( \displaystyle +b \), \( \displaystyle 2.(+a).(+b)=+2ab \)
The square of the second , the square of \( \displaystyle +b \) makes \( \displaystyle +b² \)
Lets combine them ;
\( \displaystyle (a+b)²=a²+2ab+b² \)
\( \displaystyle (a-b)² \)
The square of the first, the square of \( \displaystyle +a \) , makes \( \displaystyle +a² \)
Twice as much as the value of the result of the multiplication of the first and second term, the first \( \displaystyle +a \) , second \( \displaystyle -b \), \( \displaystyle 2.(+a).(-b)=-2ab \)
The square of the second , the square of \( \displaystyle -b \) makes \( \displaystyle +b² \)
Lets combine them ;
\( \displaystyle (a-b)²=a²-2ab+b² \)
Let’s write down a few examples;
\( \displaystyle (x+5)² \)
\( \displaystyle +x² \) , the square of the first term
\( \displaystyle +10x \) , Twice the multiplication of the first and second term
\( \displaystyle +25 \) , the square of the second term
\( \displaystyle x²+10x+25 \)
\( \displaystyle (x-5)² \)
\( \displaystyle +x² \) , the square of the first term
\( \displaystyle -10x \) , Twice the multiplication of the first and second term
\( \displaystyle +25 \) , the square of the second term
\( \displaystyle (3x-4)² \)
\( \displaystyle +9x² \) , the square of the first
\( \displaystyle -24x \) , Twice the multiplication of the first and second term
\( \displaystyle +16 \) , the square of the second
\( \displaystyle (3x-4)²=9x²-24x+16 \)
\( \displaystyle (-x+3)² \)
\( \displaystyle +x² \) , the square of the first
\( \displaystyle -6x \) , Twice the multiplication of the first and second term
\( \displaystyle +9 \) , the square of the second
\( \displaystyle (-x+3)²=x²-6x+9 \)
If you memorize by placing the notations first, you will make mistakes in these types of identities. You need to know how to separate the terms, notations will come after multiplication of the terms.
is a wrong way of writing it down !
Why do we call these identities complete square identities?
Because the identities we have found can be shown with a complete square model. We know that the lengths of the sides of a square are the same.
We need to find the area of a square with edge lengths of \( \displaystyle a+b \)
We need to find the area of a square with edge lengths of \( \displaystyle a-b \)
Because the identities we have found can be shown with a complete square model. We know that the lengths of the sides of a square are the same.
We need to find the area of a square with edge lengths of \( \displaystyle a+b \)
We need to find the area of a square with edge lengths of \( \displaystyle a-b \)
Sum of two squares and difference identities
Identity of \( \displaystyle a²+b² \)
The sum of two squares is an identity. The sum of the square of \( \displaystyle a \) + the square of \( \displaystyle b \).
We had said \( \displaystyle (a+b)²=a²+2ab+b² \), what we need is \( \displaystyle a²+b² \) . If we can add \( \displaystyle -2ab \) to either side of the identity we can obtain, \( \displaystyle a²+b² \) .
\( \displaystyle (a+b)²-2ab=a²+2ab+b²-2ab \)
\( \displaystyle -2ab \) and \( \displaystyle +2ab \) are simplified ;
\( \displaystyle (a+b)²-2ab=a²+b² \)
In a similar way;
\( \displaystyle (a-b)²=a²-2ab+b² \)
we can find it from this identity. Let’s add a \( \displaystyle +2ab \) on either side of the identity.
\( \displaystyle (a-b)²+2ab=a²-2ab+b²+2ab \)
\( \displaystyle -2ab \) and \( \displaystyle +2ab \) are simplified ;
\( \displaystyle (a-b)²+2ab=a²+b² \)
identity of \( \displaystyle a²-b² \)
We cut an area that equals to \( \displaystyle b² \)
The area is the total of these two rectangles.
\( \displaystyle a.(a-b)+b.(a-b) \)
Let's place the common multiplier \( \displaystyle (a-b) \)
within parenthese. Because it is multiplied with both \( \displaystyle a \) and \( \displaystyle b \) it is the common multiplier.
\( \displaystyle a.(a-b)+b.(a-b)=(a+b).(a-b) \)
So;
\( \displaystyle a²-b²=(a+b).(a-b) \)
It is now \( \displaystyle a²-b²=(a-b).(a+b) \) let’s find the same result by calculating backwards this time;
\( \displaystyle +ab \) is simplified with \( \displaystyle -ab \)
\( \displaystyle (a+b).(a-b)=a²-b² \)
As you can see, the same result is achieved.
Examples;
x and x multiplied is x²
+5 and ,-5 multiplied makes -25.
\( \displaystyle x²-25=(x-5).(x+5) \)
\( \displaystyle 16-a²=(4-a).(4+a) \)
\( \displaystyle 4x²-36=(2x-6).(2x+6) \)
\( \displaystyle 16x²-25y²=(4x-5y).(4x+5y) \)
Wrong Factorization!
You cannot separate it as following \( \displaystyle (4x-9). (4x+4)\), The coefficients must be the same, with a different of two squares, ( like +6 and -6 ) . For more information you can look at the topic of separating factors.
The sum of two squares is an identity. The sum of the square of \( \displaystyle a \) + the square of \( \displaystyle b \).
We had said \( \displaystyle (a+b)²=a²+2ab+b² \), what we need is \( \displaystyle a²+b² \) . If we can add \( \displaystyle -2ab \) to either side of the identity we can obtain, \( \displaystyle a²+b² \) .
\( \displaystyle (a+b)²-2ab=a²+2ab+b²-2ab \)
\( \displaystyle -2ab \) and \( \displaystyle +2ab \) are simplified ;
\( \displaystyle (a+b)²-2ab=a²+b² \)
\( \displaystyle (a-b)²=a²-2ab+b² \)
\( \displaystyle (a-b)²+2ab=a²-2ab+b²+2ab \)
\( \displaystyle -2ab \) and \( \displaystyle +2ab \) are simplified ;
\( \displaystyle (a-b)²+2ab=a²+b² \)
identity of \( \displaystyle a²-b² \)
We cut an area that equals to \( \displaystyle b² \)
The area is the total of these two rectangles.
\( \displaystyle a.(a-b)+b.(a-b) \)
Let's place the common multiplier \( \displaystyle (a-b) \)
within parenthese. Because it is multiplied with both \( \displaystyle a \) and \( \displaystyle b \) it is the common multiplier.
\( \displaystyle a.(a-b)+b.(a-b)=(a+b).(a-b) \)
So;
\( \displaystyle a²-b²=(a+b).(a-b) \)
It is now \( \displaystyle a²-b²=(a-b).(a+b) \) let’s find the same result by calculating backwards this time;
\( \displaystyle +ab \) is simplified with \( \displaystyle -ab \)
\( \displaystyle (a+b).(a-b)=a²-b² \)
As you can see, the same result is achieved.
Examples;
x and x multiplied is x²
+5 and ,-5 multiplied makes -25.
\( \displaystyle x²-25=(x-5).(x+5) \)
\( \displaystyle 16-a²=(4-a).(4+a) \)
\( \displaystyle 4x²-36=(2x-6).(2x+6) \)
\( \displaystyle 16x²-25y²=(4x-5y).(4x+5y) \)
Wrong Factorization!
You cannot separate it as following \( \displaystyle (4x-9). (4x+4)\), The coefficients must be the same, with a different of two squares, ( like +6 and -6 ) . For more information you can look at the topic of separating factors.
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