If `sqrt(x)+sqrt(x-sqrt(1-x))=1` , then show that `x=(16)/(25)` .

Answer

To solve the equation \( \sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1 \) and show that \( x = \frac{16}{25} \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \sqrt{x} + \sqrt{x - \sqrt{1 - x}} = 1 \]
We can rearrange this to isolate one of the square roots:
\[ \sqrt{x - \sqrt{1 - x}} = 1 - \sqrt{x} \]

**Hint:** Isolate one of the square roots to make squaring easier.

### Step 2: Squaring Both Sides
Next, we square both sides of the equation:
\[ \left(\sqrt{x - \sqrt{1 - x}}\right)^2 = (1 - \sqrt{x})^2 \]
This simplifies to:
\[ x - \sqrt{1 - x} = 1 - 2\sqrt{x} + x \]

**Hint:** Remember that squaring both sides will eliminate the square root but may introduce extraneous solutions.

### Step 3: Simplifying the Equation
Now, we simplify the equation:
\[ x - \sqrt{1 - x} = 1 - 2\sqrt{x} + x \]
Subtract \( x \) from both sides:
\[ -\sqrt{1 - x} = 1 - 2\sqrt{x} \]
Now, we can rearrange it to:
\[ \sqrt{1 - x} = 2\sqrt{x} - 1 \]

**Hint:** Isolate the remaining square root for the next squaring step.

### Step 4: Squaring Again
We square both sides again:
\[ (\sqrt{1 - x})^2 = (2\sqrt{x} - 1)^2 \]
This gives us:
\[ 1 - x = 4x - 4\sqrt{x} + 1 \]
Subtracting 1 from both sides results in:
\[ -x = 4x - 4\sqrt{x} \]
Rearranging gives:
\[ 5x = 4\sqrt{x} \]

**Hint:** Keep track of the terms when moving them from one side to another.

### Step 5: Isolate the Square Root
Now, we isolate the square root:
\[ \sqrt{x} = \frac{5x}{4} \]

**Hint:** This equation can be squared again to eliminate the square root.

### Step 6: Squaring Once More
Square both sides:
\[ x = \left(\frac{5x}{4}\right)^2 \]
This simplifies to:
\[ x = \frac{25x^2}{16} \]
Multiplying both sides by 16 to eliminate the fraction:
\[ 16x = 25x^2 \]

**Hint:** Rearranging this will help you form a standard quadratic equation.

### Step 7: Rearranging to Form a Quadratic Equation
Rearranging gives us:
\[ 25x^2 - 16x = 0 \]
Factoring out \( x \):
\[ x(25x - 16) = 0 \]

**Hint:** Set each factor to zero to find possible solutions.

### Step 8: Solving for \( x \)
The solutions are:
1. \( x = 0 \)
2. \( 25x - 16 = 0 \) which gives \( x = \frac{16}{25} \)

**Hint:** Check if both solutions satisfy the original equation.

### Conclusion
The only valid solution that satisfies the original equation is:
\[ x = \frac{16}{25} \]

Thus, we have shown that \( x = \frac{16}{25} \).