If - - are the roots of equation x2-4x-1-0- then the value of - 1-3- 1-3 is

The correct option is A 1 α =2+√(5) β =2 √(5) Consider x=√(2+√(5))+√(2 √(5)) ⇒ nbsp; ( x √(2+√(5)) √(2 √(5)) )=0 ⇒ nbsp; x^3 ...

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Standard XII

Mathematics

Quadratic Formula for Finding Roots

If α, β are...

Question

If $α, β$ are the roots of equation $x^{2} - 4 x - 1 = 0$ , then the value of $(α^{1 / 3} + β^{1 / 3})$ is

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Solution

The correct option is A $1$
$α = 2 + \sqrt{5}$
$β = 2 - \sqrt{5}$
Consider
$x = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$
$\Rightarrow (x - \sqrt[3]{2 + \sqrt{5}} - \sqrt[3]{2 - \sqrt{5}}) = 0$
$\Rightarrow x^{3} - (2 + \sqrt{5}) - (2 - \sqrt{5}) = 3 x (- 1)$
$\Rightarrow x^{3} - 4 = - 3 x \Rightarrow x^{3} + 3 x - 4 = 0$
$(x - 1) (x^{2} + x + 4) = 0$
Hence, $x = 1$ .

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If α,β are the roots of equation x2−4x−1=0, then the value of (α1/3+β1/3) is

The correct option is A 1α=2+√5β=2−√5Considerx=3√2+√5+3√2−√5⇒(x−3√2+√5−3√2−√5)=0⇒x3−(2+√5)−(2−√5)=3x(−1)⇒x3−4=−3x⇒x3+3x−4=0(x−1)(x2+x+4)=0Hence, x=1.

If $α, β$ are the roots of equation $x^{2} - 4 x - 1 = 0$ , then the value of $(α^{1 / 3} + β^{1 / 3})$ is