Ex 4.3, 4

Transcript

Ex 4.3, 4 Is this an identity? (𝑎+𝑏−𝑐)^2+(𝑎−𝑏+𝑐)^2+(𝑎−𝑏−𝑐)^2=2𝑎^2+2𝑏^2+2𝑐^2. We expand the left side individually Since (𝑎+𝑏+𝑐)^2=𝑎^2+𝑏^2+𝑐^2+2𝑎𝑏+2𝑏𝑐+2𝑎𝑐 Now, (𝒂+𝒃−𝒄)^𝟐= 𝑎^2+𝑏^2+〖(−𝑐)〗^2+2𝑎𝑏+2𝑏(−𝑐)+2𝑎(−𝑐) = 𝒂^𝟐+𝒃^𝟐+𝒄^𝟐+𝟐𝒂𝒃−𝟐𝒃𝒄−𝟐𝒂𝒄 (𝒂−𝒃+𝒄)^𝟐= 𝑎^2+〖(−𝑏)〗^2+ 𝑐^2+2𝑎(−𝑏)+2(−𝑏)𝑐+2𝑎𝑐 = 𝒂^𝟐+𝒃^𝟐+𝒄^𝟐−𝟐𝒂𝒃−𝟐𝒃𝒄+𝟐𝒂𝒄 (𝒂−𝒃+𝒄)^𝟐= 𝑎^2+〖(−𝑏)〗^2+ 〖(−𝑐)〗^2+2𝑎(−𝑏)+2(−𝑏)(−𝑐)+2𝑎(−𝑐) = 𝒂^𝟐+𝒃^𝟐+𝒄^𝟐−𝟐𝒂𝒃+𝟐𝒃𝒄−𝟐𝒂𝒄 Solving LHS (𝑎+𝑏−𝑐)^2+(𝑎−𝑏+𝑐)^2+(𝑎−𝑏−𝑐)^2 = 𝑎^2+𝑏^2+𝑐^2+2𝑎𝑏−2𝑏𝑐−2𝑎 + 𝑎^2+𝑏^2+𝑐^2−2𝑎𝑏−2𝑏𝑐+2𝑎𝑐 + 𝑎^2+𝑏^2+𝑐^2−2𝑎𝑏+2𝑏𝑐−2𝑎𝑐 = 3𝒂^𝟐+𝟑𝒃^𝟐+𝟑𝒄^𝟐−𝟐𝒂𝒃−𝟐𝒃𝒄−𝟐𝒂𝒄 Since this is not equal to Right side It is not an identity