Question

How does (a-b)2 become (a+b)2-4ab?

Answer 1

(a-b)²

= (a-b)(a-b)

= a² -ab - ab + b²

so

( a-b)² = a² - 2ab + b²

          = a² + b² - 2 ab + 2 ab - 2 ab    [ add 2 ab and subtract 2 ab]

          = a² + b² + 2ab - 4ab

We know that ( a+b)² = a² + b² + 2ab

        = (a+b)² - 4 ab

Answer 2

$\underline{\bold{To\:prove:-}}$

${(a - b)}^{2} = {(a + b)}^{2} - 4ab$

$\underline{\bold{Proof:-}}$

${(a - b)}^{2} \\ \\ = (a - b)(a - b) \\ \\ = a(a - b) - b(a - b) \\ \\ = {a}^{2} - ab - ab + {b}^{2} \\ \\ = {a}^{2} + {b}^{2} - 2ab \\ \\ \bold{On \: adding \: and \: subtracting \: 2ab} \\ \\ = {a}^{2} + {b}^{2} - 2ab + 2ab - 2ab \\ \\ = {a}^{2} + 2ab + {b}^{2} - 4ab \\ \\ = {a}^{2} + ab + ab + {b}^{2} - 4ab \\ \\ = a(a + b) + b(a + b) - 4ab \\ \\ = (a + b)(a + b) - 4ab \\ \\ = {(a + b)}^{2} - 4ab$

Hence proved