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

• MHB
• anemone
In summary, to solve equations involving square roots, the general strategy is to isolate the square root expression and then square both sides of the equation. For equations with nested square roots, the same strategy can be applied by isolating the inner square root and then squaring both sides. There can be extraneous solutions in these types of equations, and there may be restrictions on the variable. A shortcut method can also be used, but it may result in extraneous solutions.
anemone
Gold Member
MHB
POTW Director
Solve the equation $\sqrt{1+\sqrt{1-x^2}}(\sqrt{(1+x)^3}-\sqrt{(1-x)^3}=2+\sqrt{1-x^2}$.

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 $x = \pm\frac{1}{\sqrt{2}}$.

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

Thankyou, anemone - of course you're right!

1. What is the equation for $\sqrt{1+\sqrt{1-x^2}}$?

The equation is a nested square root function, where the inner square root is $1-x^2$ and the outer square root is $\sqrt{1+\text{inner square root}}$.

2. How do I solve for $x$ in $\sqrt{1+\sqrt{1-x^2}}$?

To solve for $x$, you will need to use the inverse operations of the nested square root function. First, square both sides of the equation to eliminate the outer square root. Then, subtract $1$ from both sides to isolate the inner square root. Finally, square both sides again to eliminate the inner square root and solve for $x$.

3. Can this equation have multiple solutions?

Yes, this equation can have multiple solutions. In fact, it can have up to four solutions depending on the value of $x$. This is because the inner square root can have both positive and negative solutions, which will result in different solutions for the entire equation.

4. What is the domain of this equation?

The domain of this equation is all real numbers except for values of $x$ that would result in a negative number under the inner square root. This is because taking the square root of a negative number is undefined in the real number system.

5. Can this equation be solved using a calculator?

Yes, this equation can be solved using a calculator. However, it is important to note that some calculators may not be able to handle nested square root functions. In this case, it may be necessary to use a graphing calculator or an online calculator that can handle more complex equations.

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