Solve the equation for x : ` 4x^2 - 4ax + (a^2 -b^2) = 0.`

Answer

To solve the quadratic equation \( 4x^2 - 4ax + (a^2 - b^2) = 0 \), we will follow these steps:

### Step 1: Identify coefficients
The general form of a quadratic equation is \( Ax^2 + Bx + C = 0 \). Here, we can identify:
- \( A = 4 \)
- \( B = -4a \)
- \( C = a^2 - b^2 \)

### Step 2: Apply the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting the values of \( A \), \( B \), and \( C \) into the formula:
\[ x = \frac{-(-4a) \pm \sqrt{(-4a)^2 - 4 \cdot 4 \cdot (a^2 - b^2)}}{2 \cdot 4} \]

### Step 3: Simplify the expression
Calculating \( B^2 \):
\[ (-4a)^2 = 16a^2 \]
Calculating \( 4AC \):
\[ 4 \cdot 4 \cdot (a^2 - b^2) = 16(a^2 - b^2) = 16a^2 - 16b^2 \]
Now substituting these back into the formula:
\[ x = \frac{4a \pm \sqrt{16a^2 - (16a^2 - 16b^2)}}{8} \]
This simplifies to:
\[ x = \frac{4a \pm \sqrt{16b^2}}{8} \]

### Step 4: Further simplify the square root
Calculating the square root:
\[ \sqrt{16b^2} = 4b \]
Substituting this back into the equation:
\[ x = \frac{4a \pm 4b}{8} \]

### Step 5: Factor out the common terms
We can factor out \( 4 \) from the numerator:
\[ x = \frac{4(a \pm b)}{8} = \frac{a \pm b}{2} \]

### Final Result
Thus, the solutions for \( x \) are:
\[ x_1 = \frac{a + b}{2}, \quad x_2 = \frac{a - b}{2} \]