if m^3 +2p^3 =3mn, a^3 +b^3 =p^3 and a^2 +b^2 =n then prove that a+b=m.


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Question Number 70103 by Shamim last updated on 01/Oct/19

$if m^{3} + {2 p}^{3} = 3 mn, a^{3} + b^{3} = p^{3} and$ $a^{2} + b^{2} = n then prove that a + b = m .$
	Answered by mind is power last updated on 01/Oct/19
	$a^3 +b^3 =(a+b)(a^2 +b^2 −ab) x=a+b ⇒x(n−ab)=p^3 ab=(1/2){(a+b)^2 −a^2 −b^2 }=(1/2)(x^2 −n) ⇒p^3 =x(n−(1/2)(x^2 −n)) ⇒2p^3 =2nx−x^3 +nx ⇒x^3 +2p^3 =3nx...E we have m solution of ..E cause m^3 +2p^3 =3mn x^3 −3nx+2p^3 =0 m^3 −3mn+2p^3 =0 ⇒(x−m)(x^2 +m^2 −mx−3n)=0 x=m or x^2 −mx+m^2 −3n=0 we have other case but depend if we solve in/R or C$
	$a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - ab)$ $x = a + b$ $\Rightarrow x (n - a b) = p^{3}$ $a b = \frac{1}{2} {{(a + b)}^{2} - a^{2} - b^{2}} = \frac{1}{2} (x^{2} - n)$ $\Rightarrow p^{3} = x (n - \frac{1}{2} (x^{2} - n))$ $\Rightarrow 2 p^{3} = 2 n x - x^{3} + n x$ $\Rightarrow x^{3} + 2 p^{3} = 3 n x . . . E$ $w e h a v e m s o l u t i o n o f . . E$ $c a u s e m^{3} + 2 p^{3} = 3 m n$ $x^{3} - 3 n x + 2 p^{3} = 0$ $m^{3} - 3 m n + 2 p^{3} = 0$ $\Rightarrow (x - m) (x^{2} + m^{2} - m x - 3 n) = 0$ $x = m$ $o r x^{2} - m x + m^{2} - 3 n = 0$ $w e h a v e o t h e r c a s e b u t d e p e n d i f w e s o l v e i n / R o r C$
	Answered by MJS last updated on 01/Oct/19

	$m^{3} + 2 p^{3} = 3 m n \Rightarrow p^{3} = \frac{m}{2} (3 n - m^{2})$ $(1) a + b = x$ $(2) a^{2} + b^{2} = n$ $(3) a^{3} + b^{3} = \frac{m}{2} (3 n - m^{2})$ $put a = α - \sqrt{β} \land b = α + \sqrt{β}$ $(1) 2 α = x$ $(2) 2 α^{2} + 2 β = n$ $(3) 2 α^{3} + 6 α β = \frac{m}{2} (3 n - m^{2})$ $(1) \Rightarrow α = \frac{x}{2}$ $(2) \Rightarrow β = \frac{n}{2} - \frac{x^{2}}{4}$ $(3) \Rightarrow β = \frac{m n}{2 x} - \frac{m^{3}}{6 x} - \frac{x^{2}}{12}$ $\Rightarrow$ $\frac{n}{2} - \frac{x^{2}}{4} = \frac{m n}{2 x} - \frac{m^{3}}{6 x} - \frac{x^{2}}{12}$ $\Leftrightarrow$ $x^{3} - 3 n x - m (m^{2} - 3 n) = 0$ $try x = \pm m \lor x = \pm (m^{2} - 3 n) \lor x = \pm m (m^{2} - 3 n)$ $\Rightarrow x_{1} = m$ $[(x - m) (x^{2} + m x + m^{2} - 3 n) = 0$ $\Rightarrow x_{2, 3} = - \frac{m}{2} \pm \frac{\sqrt{12 n - 3 m^{2}}}{2}]$ $\Rightarrow a + b = m$
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