Ex 4.3, 2

Transcript

Ex 4.3, 2 (i) Factor using suitable identities: (i) 16𝑦^2−24𝑦+9 Here, we can write 16𝑦^2=(𝟒𝒚)^𝟐 9=𝟑^𝟐 And, since there −24𝑦 i.e. negative sign, we use (a – b)2 Now, 16𝑦^2−24𝑦+9 = 16𝑦^2+9−24𝑦 = (𝟒𝒚)^𝟐+𝟑^𝟐−𝟐 × 𝟒𝒚 × 𝟑 Using (𝑎−𝑏)^2 = 𝑎^2 + 𝑏^2 – 2ab Where 𝑎 = 4𝑦, b = 3 =(𝟒𝐲−𝟑)^𝟐 Ex 4.3, 2 (ii) Factor using suitable identities: (ii) 9/4 𝑠^2+6𝑠𝑡+4𝑡^2 Here, we can write 9/4 𝑠^2=(𝟑/𝟐 𝒔)^𝟐 4𝑡^2=〖(𝟐𝒕) 〗^𝟐 And, since there +6𝑠𝑡 i.e. positive sign, we use (a + b)2 Now, 9/4 𝑠^2+6𝑠𝑡+4𝑡^2 = 9/4 𝑠^2+4𝑡^2+6𝑠𝑡 = (𝟑/𝟐 𝐬)^𝟐+(𝟐𝐭)^𝟐+𝟐 ×𝟑/𝟐 𝐬 × 𝟐𝒕 = (𝟑/𝟐 𝒔+𝟐𝒕)^𝟐 Using (𝑎+𝑏)^2 = 𝑎^2 + 𝑏^2 + 2ab Where 𝑎 = 3/2 𝑠, b = 2𝑡 Ex 4.3, 2 (iii) Factor using suitable identities: (iii) 𝑚^2/9+𝑚𝑘/3+𝑘^2/4+3𝑛𝑘+2𝑚𝑛+9𝑛^2 Here, There are 3 square terms, so we use (a + b + c)2 Our square terms would be 𝒎^𝟐/𝟗, 𝒌^𝟐/𝟒 , 𝟗𝐧^𝟐 Now, 𝑚^2/9+𝑚𝑘/3+𝑘^2/4+3𝑛𝑘+2𝑚𝑛+9𝑛^2 = 𝑚^2/9+𝑘^2/4+9𝑛^2+𝑚𝑘/3+3𝑛𝑘+2𝑚𝑛 = (𝐦/𝟑)^𝟐+(𝐤/𝟐)^𝟐+(𝟑𝒏)^𝟐+𝟐 ×𝒎/𝟑 ×𝒌/𝟐+𝟐 ×𝒌/𝟐 × 𝟑𝒏+𝟐 ×𝒎/𝟑 × 𝟑𝒏 Using (𝑎+𝑏+𝑐)^2=𝑎^2+𝑏^2+𝑐^2+2𝑎𝑏+2𝑏𝑐+2𝑎𝑐 Putting 𝑎 = m/3, 𝑏 = k/2 & 𝑐 = 3𝑛 = (𝒎/𝟑+𝒌/𝟐+𝟑𝒏)^𝟐 Ex 4.3, 2 (iv) Factor using suitable identities: (iv) 𝑝^2/16−2+16/𝑝^2 Here, we can write 𝑝^2/16=(𝒑/𝟒)^𝟐 16/𝑝^2 =(𝟒/𝒑)^𝟐 And, since there −2 i.e. negative sign, we use (a – b)2 Now, 𝑝^2/16−2+16/𝑝^2 = 𝑝^2/16+16/𝑝^2 −2 = (𝒑/𝟒)^𝟐+(𝟒/𝒑)^𝟐−𝟐 ×𝒑/𝟒 ×𝟒/𝒑 = (𝒑/𝟒−𝟒/𝒑)^𝟐 Using (𝑎−𝑏)^2 = 𝑎^2 + 𝑏^2 – 2ab Where 𝑎 = 𝑝/4, b = 4/𝑝 Ex 4.3, 2 (v) Factor using suitable identities: (v) 9𝑎^2+4𝑏^2+𝑐^2−12𝑎𝑏+6𝑎𝑐−4𝑏𝑐 Here, There are 3 square terms, so we use (a + b + c)2 Our square terms would be 𝟗𝒂^𝟐, 𝟒𝒃^𝟐 , 𝒄^𝟐 But, we have negative terms like −𝟏𝟐𝒂𝒃 and −𝟒𝒂𝒃 Since both negative term has 𝒃, it is a negative term So, we write 𝟒𝒃^𝟐=(−𝟐𝒃)^𝟐 Now, 9𝑎^2+4𝑏^2+𝑐^2−12𝑎𝑏+6𝑎𝑐−4𝑏𝑐 = (𝟑𝐚)^𝟐+(−𝟐𝐛)^𝟐+𝒄^𝟐+𝟐 × (𝟑𝒂)×(−𝟐𝒃) + 𝟐 × (𝟑𝒂) × 𝒄+𝟐 ×(−𝟐𝒃) × 𝒄 Using (𝑎+𝑏+𝑐)^2=𝑎^2+𝑏^2+𝑐^2+2𝑎𝑏+2𝑏𝑐+2𝑎𝑐 Putting 𝑎 = 3𝑎, 𝑏 = −2𝑏 & 𝑐 = 𝑐 = (3𝑎+(−2𝑏)+𝑐)^2 = (𝟑𝒂−𝟐𝒃+𝒄)^𝟐