KnowledgeBoat Logo
|

Mathematics

If x=m+13+m−13m+13−m−13x = \dfrac{\sqrt[3]{m + 1} + \sqrt[3]{m - 1}}{\sqrt[3]{m + 1} - \sqrt[3]{m - 1}}, prove that : x3 - 3x2m + 3x - m = 0.

Ratio Proportion

2 Likes

Answer

Given,

⇒x=m+13+m−13m+13−m−13\Rightarrow x = \dfrac{\sqrt[3]{m + 1} + \sqrt[3]{m - 1}}{\sqrt[3]{m + 1} - \sqrt[3]{m - 1}}

Applying componendo and dividendo, we get :

⇒x+1x−1=m+13+m−13+m+13−m−13m+13+m−13−(m+13−m−13)⇒x+1x−1=m+13+m−13+m+13−m−13m+13+m−13−m+13+m−13⇒x+1x−1=2m+132m−13⇒x+1x−1=m+13m−13\Rightarrow \dfrac{x + 1}{x - 1} = \dfrac{\sqrt[3]{m + 1} + \sqrt[3]{m - 1} + \sqrt[3]{m + 1} - \sqrt[3]{m - 1}}{\sqrt[3]{m + 1} + \sqrt[3]{m - 1} - (\sqrt[3]{m + 1} - \sqrt[3]{m - 1})} \\[1em] \Rightarrow \dfrac{x + 1}{x - 1} = \dfrac{\sqrt[3]{m + 1} + \sqrt[3]{m - 1} + \sqrt[3]{m + 1} - \sqrt[3]{m - 1}}{\sqrt[3]{m + 1} + \sqrt[3]{m - 1} - \sqrt[3]{m + 1} + \sqrt[3]{m - 1}} \\[1em] \Rightarrow \dfrac{x + 1}{x - 1} = \dfrac{2\sqrt[3]{m + 1}}{2\sqrt[3]{m - 1}} \\[1em] \Rightarrow \dfrac{x + 1}{x - 1} = \dfrac{\sqrt[3]{m + 1}}{\sqrt[3]{m - 1}}

Cubing both sides, we get :

⇒(x+1x−1)3=m+1m−1⇒(x+1)3(m−1)=(x−1)3(m+1)⇒(x3+3x2+3x+1)(m−1)=(x3−3x2+3x−1)(m+1)⇒mx3+3mx2+3mx+m−x3−3x2−3x−1=mx3−3mx2+3mx−m+x3−3x2+3x−1⇒mx3+3mx2+3mx+m−x3−3x2−3x−1−(mx3−3mx2+3mx−m+x3−3x2+3x−1)=0⇒mx3−mx3+3mx2+3mx2+3mx−3mx+m+m−x3−x3−3x2+3x2−3x−3x−1+1=0⇒6mx2+2m−2x3−6x=0⇒2(3mx2+m−x3−3x)=0⇒3mx2+m−x3−3x=0⇒x3−3mx2+3x−m=0.\Rightarrow \Big(\dfrac{x + 1}{x - 1}\Big)^3 = \dfrac{m + 1}{m - 1} \\[1em] \Rightarrow (x + 1)^3 (m - 1) = (x - 1)^3 (m + 1) \\[1em] \Rightarrow (x^3 + 3x^2 + 3x + 1)(m - 1) = (x^3 - 3x^2 + 3x - 1)(m + 1) \\[1em] \Rightarrow m x^3 + 3m x^2 + 3m x + m - x^3 - 3x^2 - 3x - 1 = m x^3 - 3m x^2 + 3m x - m + x^3 - 3x^2 + 3x - 1 \\[1em] \Rightarrow m x^3 + 3m x^2 + 3m x + m - x^3 - 3x^2 - 3x - 1 -( m x^3 - 3m x^2 + 3m x - m + x^3 - 3x^2 + 3x - 1) = 0 \\[1em] \Rightarrow m x^3 - m x^3 + 3m x^2 + 3m x^2 + 3m x - 3m x + m + m - x^3 - x^3 - 3x^2 + 3x^2 - 3x - 3x - 1 + 1 = 0 \\[1em] \Rightarrow 6m x^2 + 2m - 2x^3 - 6x = 0 \\[1em] \Rightarrow 2(3mx^2 + m - x^3 - 3x) = 0 \\[1em] \Rightarrow 3mx^2 + m - x^3 - 3x = 0 \\[1em] \Rightarrow x^3 - 3mx^2 + 3x - m = 0.

Hence, proved that x3 - 3x2m + 3x - m = 0.

Answered By

2 Likes

Related Questions