Algebraic Identities

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Introduction to Algebraic Identities

Let us say that we are dealing with two equations. The two equations are:

\({\,\,5x \,– 3 = 12}\)

\({10x \,– 6 = 24}\)

If you solve both equations separately, you will see that the value of \({x}\) is 3 in both cases. If you write the equations in the form \({ax \,– b = c,}\) then you will see that the two equations are:

\({ax\, – b = c}\)

\({2(ax \,– b) = 2c}\)

What you did there was notice a pattern. Allowing us to observe patterns is how algebra appears in our daily lives.

Algebraic Identities Explained Visually

What is an identity in math?

  • An identity is an equation that is always true regardless of the values assigned to the variables.
  • Identity means that the left-hand side of the equation is identical to the right-hand side, for all values of the variables.

\(\begin{align} \text{E.g. } (a + b)^2 = a^2 + 2ab + b^2 \text{ is true for all values of } a \text{ and } b\end{align}\)

Most equations in math work only for certain specific values.

\(\begin{align} \text{E.g. } 4x + 5 = 17 \text{ is true only if } x = 3\end{align}\)

Because identities are always true regardless of the value of the variables, they are very useful.

List of the most important and useful identities

\[\begin{align} \boxed{(a + b)^2   =   a^2 + 2ab + b^2} \end{align}\]

\[\begin{align} \boxed{(a - b)^2   =   a^2 - 2ab + b^2} \end{align}\]

\[\begin{align} \boxed{(a + b)(a − b) = a^2 − b^2} \end{align}\]

\[\begin{align} \boxed{(𝑥 + a) (x+ b) = x^2 + x(a+b) + ab} \end{align}\]

Geogebra Simulation:

Page title Algebraic Identities_Ranganath
Simulation name ( a + b )^2
Concept name Algebraic identity
Brief description

This Simulation can show that identities hold true always by taking an example of ( a + b )^2. Here we take different values of a and b and show both sides of the identity are equal. The user controls the value of a and b. 

Dont memorise! You can visualise these identities

Visualise \(\begin{align} (a + b)^2 \end{align}\)

Think of \(\begin{align} (a + b) \end{align}\) as a length. 
So, \(\begin{align} (a + b)^2 = (a + b) × (a + b) \end{align}\) will be length squared. 
This can be visualised as an area of a square with length \(\begin{align} (a + b) \end{align}\).
Creating a square with length \(\begin{align} (a + b) \end{align}\) gives us area which has 4 parts as shown here:

So, \(\begin{align} (a + b)^2 \end{align}\) is simply the area of this square. Adding the 4 parts we get:

\(\begin{align} (a + b)^2   =   a^2 + ab + ab + b^2 \end{align}\)

\(\begin{align} (a + b)^2   =   a^2 + 2ab + b^2 \end{align}\)

Now \((a - b)^2\) can be done in a similar way. Try it out yourself.(Hint: Start with a square of side length \(a\).)

Visualise \(\begin{align} (a + b) (a − b) \end{align}\)

To visualise this identity, we start once again by visualising a rectangle. 

The rectangle will have a length \(\begin{align} (a + b) \end{align}\) and breadth \(\begin{align} (a - b) \end{align}\). 

The rectangle is shown by the blue and yellow areas together.

Now visualise the yellow vertical strip moving to the bottom of the blue area and aligning horizontally as shown below. 

Since we have simply moved it without stretching it, the area remains unchanged.

When viewed this way, we see that the area is now equal to \(\begin{align} a^2 - b^2 \end{align}\).
\[\begin{align} (a + b) (a − b)  = a^2 - b^2 \end{align}\] 

Video Brief:

Page Titie Algebraic Identities_Ranganath
Video Name Proof of Algebraic Identities
Concept name Algebraic Identities
Target audience Grade 8
Video script The video shows a student trying to learn the proof of the identities by-heart. Then his friend tells him to look at the video of the geometric proof of the identities and shows it to him. The proof could be ; For (a+b)^2, it can show a big square of length (a+b). and how the terms in the expansion are smaller squares, which fit perfectly in the big square. Like this it shows the proof of all 4 identities. 

Solved Examples

Example 1:

Using algebraic identities, find \((2x + 3y)^2\).

Solution:

This is similar to the identity \(\begin{align} (a + b)^2   =   a^2 + 2ab + b^2 \end{align}\), where \(( 'a'\) is \(2x\) and \('b'\) is \(3y)\)

\(\begin{align} (2x + 3y)^2 &= (2x)^2 + 2(2x)(3y) + (3y)^2 \\ &= 4x^2 + 12xy + 9y^2 \end{align}\).

Example 2:

Using identities, evaluate \( 297 \times 303 \).

Solution:

The above form can be written as \(( 300 - 3 ) \times ( 300 + 3 )\). 

This is based on the formula : \(\begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}\), where \(( 'a' = 300 \) and \( 'b' = 3 )\).

Putting the values in the above identity, we get: 

 \(\begin{align} (300 - 3)(300 + 3) &= 300^2 − 3^2 \\ &= 90000 - 9 \\ &= 89991 \end{align}\)

Example 3:

Expand \(( pqr + 2 )( pqr - 5 )\).

Solution:

This is based on the formula \(\begin{align}(𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align}\),

where  \(( 'x' = pqr \) and \( 'a' = 2 \) and \( 'b' = -5)\).

So, substituting in the above equation:

\(\begin{align} ( pqr + 2 )( pqr - 5 ) &= (pqr)^2 + pqr(2 + (-5)) + (2)(-5) \\ &= p^2q^2r^2 -3pqr -10 \end{align}\)

Example 4:

Simplify \( ( 7x +4y )^2 + ( 7x - 4y )^2\)

Solution:

For this, we need to use the following formulas:

\(\begin{align} (a + b)^2   =   a^2 + 2ab + b^2 \end{align}\)

\(\begin{align} (a - b)^2    =   a^2  - 2ab + b^2 \end{align}\)

Here  \(( 'a' = 7x \) and \( 'b' = 4y )\).

So, 

\(\begin{align} ( 7x +4y )^2 + ( 7x - 4y )^2 &= ( 49x^2 + 56xy + 16y^2) + (49x^2 - 56xy + 16y^2) \\ &= 98x^2 + 32y^2 \end{align}\)

Practice Questions

1. Expand the following expressions.

  •  \((4x - 3y)^2\)
  •  \((4x - 5)(4x +1)\)

2.  Simplfy.

  • \((3x + 5)^2 - (3x - 5)^2\)
  • \((p^2 - q^2p)^2 + 2p^3q^2\)

3. Using identities, evaluate the following:

  • \( 398 \times 402 \)
  • \( 204^2 \)
  • \( 998^2 \)

Activity

Activity name: Sub_AlgebraicIdentities_Activity
Item name 1 Sub_AlgebraicIdentities_Item1
Item name 2 Sub_AlgebraicIdentities_Item2
Item name 3 Sub_AlgebraicIdentities_Item3
Item name 4 Sub_AlgebraicIdentities_Item4
Item name 5 Sub_AlgebraicIdentities_Item5

Tips and Tricks related to Algebraic Identities

  • Visualise identities as rectangles. That way even if you forget, you can quickly work out the expressions.
    E.g. \(\begin{align} (a + b)^2 \end{align}\) can be visualised as a square with sides\(\begin{align} (a + b) \end{align}\) giving four parts: \(\begin{align} a^2,\;b^2 \end{align}\) and two \(ab\).

  • Remember the factored forms rather than the simplified form. It’s difficult to remember what \(\begin{align} a^2 - b^2 \end{align}\) factorises to. But if you have the factored form \(\begin{align} (a + b) (a - b) \end{align}\), you can work out the expression even if you don’t remember it.

Frequently Asked Questions

1. What are the four algebraic identities?

Ans. The four commonly used algebraic identities are given below:

  • \(\begin{align} (a + b)^2   =   a^2 + 2ab + b^2 \end{align}\)
  • \(\begin{align} (a - b)^2   =   a^2 - 2ab + b^2 \end{align}\)
  • \(\begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}\)
  • \(\begin{align} (𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align}\)

2. How can I learn algebraic identity?

Ans.

  • Algebraic identities can be easily memorized by visualising the identities as squares or rectangles.
  • It also helps by remembering the factored forms rather than the simplified form.

3. What is the formula of an identity?

Ans.

  • An identity is an equation that is always true regardless of the value that is assigned to the variable.  
  • Equations are those which hold true for only specific value of the variable.

4. How many identities are there?

Ans.There are many identities, but the most common ones are given below:

  • \(\begin{align} (a + b)^2   =   a^2 + 2ab + b^2 \end{align}\)
  • \(\begin{align} (a - b)^2   =   a^2 - 2ab + b^2 \end{align}\)
  • \(\begin{align} (a + b)(a − b) = a^2 − b^2 \end{align}\)
  • \(\begin{align} (𝑥 + a) (x+ b) = x^2 + x(a+b) + ab \end{align}\)

5. Are all identities equations?

Ans. Yes, since identities are equation that hold true for any value of the variable.

 

  
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