The product of the roots of the equation `sqrt(5x+8)=sqrt(x^(2)-16)` is :

Options

A. `-64`
B. `-24`
C. `64`
D. `24`

Answer

To solve the equation \( \sqrt{5x + 8} = \sqrt{x^2 - 16} \) and find the product of its roots, we can follow these steps:

### Step 1: Square both sides
To eliminate the square roots, we square both sides of the equation:
\[ (\sqrt{5x + 8})^2 = (\sqrt{x^2 - 16})^2 \]
This simplifies to:
\[ 5x + 8 = x^2 - 16 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ x^2 - 16 - 5x - 8 = 0 \]
This simplifies to:
\[ x^2 - 5x - 24 = 0 \]

### Step 3: Factor the quadratic equation
Now we need to factor the quadratic equation \( x^2 - 5x - 24 = 0 \). We look for two numbers that multiply to \(-24\) and add to \(-5\). The numbers \(-8\) and \(3\) work:
\[ (x - 8)(x + 3) = 0 \]

### Step 4: Find the roots
Setting each factor to zero gives us the roots:
1. \( x - 8 = 0 \) → \( x = 8 \)
2. \( x + 3 = 0 \) → \( x = -3 \)

### Step 5: Calculate the product of the roots
The product of the roots \( r_1 \) and \( r_2 \) is:
\[ r_1 \cdot r_2 = 8 \cdot (-3) = -24 \]

Thus, the product of the roots of the equation is \(-24\).

### Final Answer:
The product of the roots is \(-24\).

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