Question 1 - Think and Reflect (Page 78) - Rewriting a2 - b2 Identity - Chapter 4 Class 9 - Exploring Algebraic Identities (Ganita Manjari I)

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Question 1 - Think and Reflect (Page 78) Try to evaluate the following using a suitable identity: (i) 35^2 Writing 35 = 30 + 5 So, we can write 〖𝟑𝟓〗^𝟐=(𝟑𝟓 −𝟓)(𝟑𝟓+𝟓)+𝟓^𝟐 =30 × 40+25 =1,200+25 =𝟏,𝟐𝟐𝟓 Using 𝑎^2=(𝑎 −𝑏)(𝑎+𝑏)+𝑏^2 Putting 𝒂 = 𝟑𝟓, 𝒃 = 𝟓 Question 1 - Think and Reflect (Page 78) Try to evaluate the following using a suitable identity: (ii) 65^2 Writing 65 = 60 + 5 So, we can write 〖𝟔𝟓〗^𝟐=(𝟔𝟓 −𝟓)(𝟔𝟓+𝟓)+𝟓^𝟐 =60 × 70+25 =4,200+25 =𝟒,𝟐𝟐𝟓 Using 𝑎^2=(𝑎 −𝑏)(𝑎+𝑏)+𝑏^2 Putting 𝒂 = 𝟔𝟓, 𝒃 = 𝟓 Question 1 - Think and Reflect (Page 78) Try to evaluate the following using a suitable identity: (iii) 85^2 Writing 85 = 80 + 5 So, we can write 〖𝟖𝟓〗^𝟐=(𝟖𝟓 −𝟓)(𝟖𝟓+𝟓)+𝟓^𝟐 =80 × 90+25 =7,200+25 =𝟕,𝟐𝟐𝟓 Using 𝑎^2=(𝑎 −𝑏)(𝑎+𝑏)+𝑏^2 Putting 𝒂 = 8𝟓, 𝒃 = 𝟓 Question 1 - Think and Reflect (Page 78) Try to evaluate the following using a suitable identity: (iv) 105^2 Writing 105 = 100 + 5 So, we can write 〖𝟏𝟎𝟓〗^𝟐=(𝟏𝟎𝟓 −𝟓)(𝟏𝟎𝟓+𝟓)+𝟓^𝟐 =100 × 110+25 =11,000+25 =𝟏𝟏,𝟎𝟐𝟓 Using 𝑎^2=(𝑎 −𝑏)(𝑎+𝑏)+𝑏^2 Putting 𝒂 = 𝟏𝟎𝟓, 𝒃 = 𝟓 Now, we are asked Do you observe any interesting pattern? We don't even need the formula if a number ends in 5! Look at 652 = 4225. The answer always ends in 25 To get the first numbers, take the first digit of your original number (for 65, it's 6) and multiply it by the next number up (which is 7). Multiplication is 6 × 7 = 42 Put them together: 4225 Let's test it on 852 The first digit is 8 Multiply by the next number (9) Multiplication is 8 × 9 = 72 Slap a 25 on the end Thus, 852 = 7225