MHB Solve Equation: $\sqrt{1+\sqrt{1-x^2}}$

[anemone]

Solve the equation $\sqrt{1+\sqrt{1-x^2}}(\sqrt{(1+x)^3}-\sqrt{(1-x)^3}=2+\sqrt{1-x^2}$.

 

[lfdahl]

Spoiler: Suggested solution

Rewriting the equation with $a = \sqrt{1+x}$ and $b = \sqrt{1-x}$:

$\sqrt{1+ab}\left ( a^3 - b^3 \right ) = 2 + ab$

$\sqrt{1+ab}\left ( a - b \right )(a^2+b^2+ab) = 2 + ab$

$\sqrt{1+ab}\left ( a - b \right ) = 1$ - because $a^2+b^2 = 2$.

Squaring yields:

$(1+ab)\left ( a^2 + b^2-2ab \right ) = 1$

or

$2(1+ab)(1-ab)= 1$

or $(ab)^2 = \frac{1}{2}$

• hence $1-x^2 =\frac{1}{2}$ or $x = \pm\frac{1}{\sqrt{2}}$.

 

[anemone]

Hi lfdahl! The correct answer is $x=\dfrac{1}{\sqrt{2}}$ only. (Smile)

 

[lfdahl]

Thankyou, anemone - of course you're right!