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 use the quadratic formula:

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

### Step 1: Identify the coefficients
In the given quadratic equation, we can identify the coefficients as follows:
- \(A = 4\) (coefficient of \(x^2\))
- \(B = -4a\) (coefficient of \(x\))
- \(C = a^2 + b^2\) (constant term)

### Step 2: Substitute the coefficients into the quadratic formula
Now, we substitute \(A\), \(B\), and \(C\) into the quadratic formula:

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

### Step 3: Simplify the expression
First, simplify the expression for \(x\):

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

Now, simplify the expression under the square root:

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

So, we have:

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

### Step 4: Simplify the square root
The square root of \(-16b^2\) can be simplified:

\[ \sqrt{-16b^2} = \sqrt{16} \cdot \sqrt{-1} \cdot |b| = 4b i \]

### Step 5: Substitute back into the equation
Now substitute this back into the equation for \(x\):

\[ x = \frac{4a \pm 4bi}{8} \]

### Step 6: Simplify the fraction
Now we can simplify the fraction:

\[ x = \frac{4a}{8} \pm \frac{4bi}{8} = \frac{a}{2} \pm \frac{bi}{2} \]

### Final Answer
Thus, the solutions for the equation \(4x^2 - 4ax + (a^2 + b^2) = 0\) are:

\[ x = \frac{a + bi}{2} \quad \text{and} \quad x = \frac{a - bi}{2} \]