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Exploring Algebraic Identities - Introduction

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An algebraic identity is an algebraic equation that is true for all values of the variables occurring in it. For example, (x+1)2=x2+2x+1(x+1)^2 = x^2 + 2x + 1 is an identity because it holds true for any value of xx.

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Algebraic identities are useful for expanding products of binomials and for factorizing algebraic expressions.

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Standard identities can be used to perform numerical calculations more efficiently without direct multiplication, such as calculating squares of large numbers like 99299^2 or products like 103×97103 \times 97.

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Identity IV, (x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab, is specifically used when the first terms of two binomials are identical but the second terms are different.

📐Formulae

(x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2

(x−y)2=x2−2xy+y2(x - y)^2 = x^2 - 2xy + y^2

x2−y2=(x+y)(x−y)x^2 - y^2 = (x + y)(x - y)

(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab

💡Examples

Problem 1:

Expand the expression (3a+4b)2(3a + 4b)^2 using a suitable identity.

Solution:

Using Identity I: (x+y)2=x2+2xy+y2(x + y)^2 = x^2 + 2xy + y^2. Here, x=3ax = 3a and y=4by = 4b. (3a+4b)2=(3a)2+2(3a)(4b)+(4b)2(3a + 4b)^2 = (3a)^2 + 2(3a)(4b) + (4b)^2 =9a2+24ab+16b2= 9a^2 + 24ab + 16b^2

Explanation:

We identify the terms corresponding to xx and yy and substitute them into the square of a binomial formula.

Problem 2:

Evaluate 105×95105 \times 95 without multiplying directly.

Solution:

We can write the numbers as (100+5)(100 + 5) and (100−5)(100 - 5). Using Identity III: (x+y)(x−y)=x2−y2(x + y)(x - y) = x^2 - y^2. Here, x=100x = 100 and y=5y = 5. 105×95=(100+5)(100−5)105 \times 95 = (100 + 5)(100 - 5) =1002−52= 100^2 - 5^2 =10000−25= 10000 - 25 10000−259975\begin{array}{r} 10000 \\ - 25 \\ \hline 9975 \end{array} Final Answer: 99759975.

Explanation:

The numbers are equidistant from 100100, allowing us to use the difference of squares identity.

Problem 3:

Find the product of (x−3)(x+5)(x - 3)(x + 5) using algebraic identities.

Solution:

Using Identity IV: (x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab. Here, a=−3a = -3 and b=5b = 5. (x−3)(x+5)=x2+(−3+5)x+(−3)(5)(x - 3)(x + 5) = x^2 + (-3 + 5)x + (-3)(5) =x2+2x−15= x^2 + 2x - 15

Explanation:

Since the first term xx is common, we use Identity IV by carefully including the negative sign for the constant aa.