Solve the following problems

If $x + y = a x^{2} + y^{2} = b^{2}$ and $x^{3} + y^{3} = c^{3}$ then show that $a^{3} + 2 c^{3} = 3 a b^{2}$

Created: 3 weeks ago | Updated: 1 week ago

Job Solution One Bank Ltd One Bank Ltd || Special Cadre Officer (03-01-2025) || 2025 গণিত

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Here,

$L . H . S = a^{3} + 2 c^{3} = (x + y)^{3} + 2 (x^{3} + y^{3})$

$= x^{3} + y^{3} + 3 x^{2} * y + 3 x * y^{2} + 2 x^{3} + 2 y^{3}$

$= 3 x^{3} + 3 y^{3} + 3 x^{2} y + 3 x y^{2}$

$= 3 (x^{3} + y^{3}) + 3 x y (x + y)$

$= 3 (x + y) (x^{2} - x y + y^{2}) + 3 x y (x + y)$

$= 3 (x + y) (x^{2} - x y + y^{2} + x y)$

$= 3 (x + y) (x^{2} + y^{2}) = 3 a b^{2}$

= R.H.S (Proved)

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