Question

please answer fast it will help me a lot​

Answer 1

Explanation

$\colon\implies{\sf{ a^2 + \dfrac{1}{4} b^2 + \dfrac{1}{9} c^2 + ab - \dfrac{1}{3} bc - \dfrac{2}{3} ca }}$

We can use this Identity to Express this Question as :-

$\colon\implies{\pmb{\sf\red{ (a + b + c)^2 = a^2+b^2+c^2+2(ab+bc+ca) }}}$

$\\ \colon\implies{\sf{ (a)^2 + \left( \dfrac{b}{2} \right)^2 + \left( \dfrac{c}{3} \right)^2 + 2(a) \left( \dfrac{1}{2} b \right) + 2 \left( \dfrac{1}{2} b \right) \left( - \dfrac{1}{3} c \right) + 2 \left( - \dfrac{1}{3} c \right) (a) }}$

$\\ \colon\implies{\sf{ (a)^2 + \left( \dfrac{1}{2} b \right)^2 + \left( - \dfrac{1}{3} c \right)^2 + 2(a) \left( \dfrac{1}{2} ab \right) +2 \left( \dfrac{1}{2} b \right) \left( - \dfrac{1}{3} c \right) + 2 \left( - \dfrac{1}{3} c \right) (a) = \left[ a + \dfrac{1}{2} b + \left( - \dfrac{1}{3} c \right) \right] ^2 }}$

$\\ \colon\implies{\sf{ (a)^2 + \left( \dfrac{1}{2} b \right)^2 + \left( - \dfrac{1}{3} c \right)^2 + 2(a) \left( \dfrac{1}{2} b \right) + 2 \left( \dfrac{1}{2} b \right) \left( - \dfrac{1}{3} c \right) + 2 \left( - \dfrac{1}{3} c \right) (a) = \left( a + \dfrac{1}{2} b - \dfrac{1}{3} c \right)^2 }}$

$\\ \colon\implies{\pmb{\sf{ \left( a + \dfrac{b}{2} - \dfrac{c}{3} \right)^2 }}}$

$\\ \maltese \ {\pmb{\underline{\sf{ More \ to \ Know : }}}} \\ \\ \circ{\boxed{\sf\large{ (a+b)^2 = a^2+b^2+2ab }}} \\ \\ \circ{\boxed{\sf\large{ (a-b)^2 = a^2+b^2-2ab }}} \\ \\ \circ{\boxed{\sf\large{ a^2-b^2 = (a+b)(a-b) }}} \\ \\ \circ{\boxed{\sf\large{ (a+b)^3=a^3+b^3+3ab(a+b) }}} \\ \\ \circ{\boxed{\sf\large{ (a-b)^3 = a^3-b^3-3ab(a-b) }}}$