Total number of values of x satisfying -3-12x-3-12x-23x is

(√(3)+1)^2x+(√(3) 1)^2x=2^3x The above equation can be re written as nbsp; nbsp; (√(3)+1)^2x+(√(3) 1)^2x=(2√(2))^2x ⇒(√(3)+12√(2))^2x+(√(3) 12√(2))^2x=1 ⇒ ( ...

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Standard XI

Mathematics

Higher Order Equations

Total number ...

Question

Total number of values of $x$ satisfying $(\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = 2^{3 x}$ is

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Solution

$(\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = 2^{3 x}$
The above equation can be re-written as
$(\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = (2 \sqrt{2})^{2 x}$
$\Rightarrow {(\frac{\sqrt{3} + 1}{2 \sqrt{2}})}^{2 x} + {(\frac{\sqrt{3} - 1}{2 \sqrt{2}})}^{2 x} = 1$
$\Rightarrow (sin 75^{\circ})^{2 x} + (cos 75^{\circ})^{2 x} = 1$
$∴ x = 1$

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Total number of values of x satisfying (√3+1)2x+(√3−1)2x=23x is

(√3+1)2x+(√3−1)2x=23x The above equation can be re-written as (√3+1)2x+(√3−1)2x=(2√2)2x ⇒(√3+12√2)2x+(√3−12√2)2x=1 ⇒(sin75∘)2x+(cos75∘)2x=1 ∴x=1

Total number of values of $x$ satisfying $(\sqrt{3} + 1)^{2 x} + (\sqrt{3} - 1)^{2 x} = 2^{3 x}$ is