Visualising Algebraic Identities

Transcript

Visualising Algebraic Identities Let’s take two numbers a & b If we add them: (a + b) We get a line of length (a + b) If we add them, & square: (a + b)2 We get a square of length (a + b) Where (a + b)2 = a2 + b2 + 2ab Let’s take an example: a = 5, b = 3 Step 1: The 1D Length Step 2: Squaring Step 3: The Total Area = Sum of the 4 smaller areasAlgebraic Proof To prove 𝑎^2+2𝑎𝑏+𝑏^2=(𝑎+𝑏)^2 Solving (𝐚+𝐛)^𝟐 (𝒂+𝒃)^𝟐=(𝑎+𝑏)(𝑎+𝑏) =𝒂(𝒂+𝒃)+𝒃(𝒂+𝒃) =𝑎 × 𝑎+𝑎 × 𝑏+𝑏 × 𝑎+𝑏 × 𝑏 =𝑎^2+𝑎𝑏+𝑏𝑎+𝑏^2 Since ab = ba =𝑎^2+𝑎𝑏+𝑎𝑏+𝑏^2 =𝒂^𝟐+𝟐𝒂𝒃+𝒃^𝟐 Hence proved