If `sqrt(4x-9) + sqrt(4x +9) = 5 +sqrt7` then what is the value of x ?

Options

A. 5
B. 7
C. 3
D. 4

Answer

To solve the equation \(\sqrt{4x-9} + \sqrt{4x+9} = 5 + \sqrt{7}\), we will follow these steps:

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square roots.

\[ (\sqrt{4x-9} + \sqrt{4x+9})^2 = (5 + \sqrt{7})^2 \]

### Step 2: Expand both sides
Now we expand both sides:

Left-hand side:
\[ (\sqrt{4x-9})^2 + 2\sqrt{(4x-9)(4x+9)} + (\sqrt{4x+9})^2 = (4x - 9) + (4x + 9) + 2\sqrt{(4x-9)(4x+9)} \]
\[ = 8x + 2\sqrt{(4x-9)(4x+9)} \]

Right-hand side:
\[ (5 + \sqrt{7})^2 = 25 + 2 \cdot 5 \cdot \sqrt{7} + 7 = 32 + 10\sqrt{7} \]

### Step 3: Set the equation
Now we have:
\[ 8x + 2\sqrt{(4x-9)(4x+9)} = 32 + 10\sqrt{7} \]

### Step 4: Isolate the square root term
Next, we isolate the square root term:
\[ 2\sqrt{(4x-9)(4x+9)} = 32 + 10\sqrt{7} - 8x \]

### Step 5: Divide by 2
Now, divide everything by 2:
\[ \sqrt{(4x-9)(4x+9)} = 16 + 5\sqrt{7} - 4x \]

### Step 6: Square both sides again
Square both sides again to eliminate the square root:
\[ (4x-9)(4x+9) = (16 + 5\sqrt{7} - 4x)^2 \]

### Step 7: Expand both sides
Left-hand side:
\[ (4x-9)(4x+9) = 16x^2 - 81 \]

Right-hand side:
\[ (16 + 5\sqrt{7} - 4x)^2 = (16 - 4x)^2 + 2(16 - 4x)(5\sqrt{7}) + (5\sqrt{7})^2 \]
\[ = (16 - 4x)^2 + 2(16)(5\sqrt{7}) - 8x(5\sqrt{7}) + 175 \]

### Step 8: Combine terms
Now combine and simplify the right-hand side:
\[ = 256 - 128x + 16x^2 + 175 + 160\sqrt{7} - 40x\sqrt{7} \]
\[ = 16x^2 - 128x + 431 + 160\sqrt{7} - 40x\sqrt{7} \]

### Step 9: Set the equation
Now we set both sides equal:
\[ 16x^2 - 81 = 16x^2 - 128x + 431 + 160\sqrt{7} - 40x\sqrt{7} \]

### Step 10: Simplify and solve for x
Now, we can cancel \(16x^2\) from both sides:
\[ -81 = -128x + 431 + 160\sqrt{7} - 40x\sqrt{7} \]
Rearranging gives:
\[ 128x = 431 + 81 + 160\sqrt{7} - 40x\sqrt{7} \]
\[ 128x = 512 + 160\sqrt{7} - 40x\sqrt{7} \]
Combining terms gives:
\[ (128 + 40\sqrt{7})x = 512 + 160\sqrt{7} \]
Now, divide by \((128 + 40\sqrt{7})\):
\[ x = \frac{512 + 160\sqrt{7}}{128 + 40\sqrt{7}} \]

### Step 11: Simplify x
This can be simplified further, but it requires rationalizing the denominator or substituting numerical values for \(\sqrt{7}\).

### Final Result
After simplification, we find:
\[ x = 4 \]