if-m-3-2p-3-3mn-a-3-b-3-p-3-and-a-2-b-2-n-then-prove-that-a-b-m-

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

\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 \dots 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

Facebook Tweet Pin