Solve each of the following equatins :
`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 the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 4 \)
- \( b = -4a \)
- \( c = a^2 - b^2 \)

### Step 2: Use the quadratic formula
The solutions for a 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 \( b^2 \):
\[ b^2 = (-4a)^2 = 16a^2 \]
Calculating \( 4ac \):
\[ 4ac = 4 \cdot 4 \cdot (a^2 - b^2) = 16(a^2 - b^2) \]
Now substituting these into the formula:
\[ x = \frac{4a \pm \sqrt{16a^2 - 16(a^2 - b^2)}}{8} \]
This simplifies to:
\[ x = \frac{4a \pm \sqrt{16a^2 - 16a^2 + 16b^2}}{8} \]
\[ x = \frac{4a \pm \sqrt{16b^2}}{8} \]
\[ x = \frac{4a \pm 4b}{8} \]

### Step 4: Factor out common terms
Factoring out the common term:
\[ x = \frac{4(a \pm b)}{8} \]
This simplifies to:
\[ x = \frac{a \pm b}{2} \]

### Step 5: Write the final solutions
Thus, the solutions for the quadratic equation are:
\[ x = \frac{a + b}{2} \quad \text{and} \quad x = \frac{a - b}{2} \]

### Summary of Solutions
The two values of \( x \) are:
1. \( x = \frac{a + b}{2} \)
2. \( x = \frac{a - b}{2} \)

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