Find the roots of the given quadratic equation .
`4x^(2) + 4bx - (a^(2) - b^(2)) = 0`

Options

A. `(a+b)/(2),(a+b)/(2)`
B. `(a-b)/(2),(a+b)/(2)`
C. `(a-b)/(2),(a-b)/(2)`
D. `(a-b)/(2),-((a+b))/(2)`

Answer

To find the roots of the quadratic equation \(4x^2 + 4bx - (a^2 - b^2) = 0\), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
In the given equation, we can identify the coefficients:
- \(a = 4\)
- \(b = 4b\)
- \(c = -(a^2 - b^2)\)

### Step 2: Substitute coefficients into the quadratic formula
Now, we substitute these values into the quadratic formula:

\[ x = \frac{-4b \pm \sqrt{(4b)^2 - 4 \cdot 4 \cdot (-(a^2 - b^2))}}{2 \cdot 4} \]

### Step 3: Simplify the expression
First, we simplify the expression under the square root:

\[ (4b)^2 = 16b^2 \]
\[ -4 \cdot 4 \cdot (-(a^2 - b^2)) = 16(a^2 - b^2) \]

Thus, the expression under the square root becomes:

\[ 16b^2 + 16(a^2 - b^2) = 16b^2 + 16a^2 - 16b^2 = 16a^2 \]

Now substituting back into the formula:

\[ x = \frac{-4b \pm \sqrt{16a^2}}{8} \]

### Step 4: Further simplify
Since \(\sqrt{16a^2} = 4|a|\), we have:

\[ x = \frac{-4b \pm 4|a|}{8} \]

### Step 5: Divide by 4
Now, we can simplify this further:

\[ x = \frac{-b \pm |a|}{2} \]

### Step 6: Write the final roots
Thus, the roots of the quadratic equation are:

\[ x_1 = \frac{-b + |a|}{2} \quad \text{and} \quad x_2 = \frac{-b - |a|}{2} \]

### Summary of the Roots
The roots of the quadratic equation \(4x^2 + 4bx - (a^2 - b^2) = 0\) are:

\[ x_1 = \frac{-b + |a|}{2}, \quad x_2 = \frac{-b - |a|}{2} \]