If the roots of the equation `(a^(2) + b^(2)) x^(2) - 2b (a + c)x + (b^(2) + c^(2))` = 0 are equal, then which one of the following is correct ?

Options

A. 2b = a + c
B. `b^(2) = ac`
C. `b + c = 2a`
D. b = ac

Answer

To determine the condition under which the roots of the quadratic equation

\[ (a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0 \]

are equal, we need to use the condition that the discriminant of the quadratic equation must be zero.

### Step 1: Identify the coefficients
In the standard form of a quadratic equation \(Ax^2 + Bx + C = 0\), we can identify:
- \(A = a^2 + b^2\)
- \(B = -2b(a + c)\)
- \(C = b^2 + c^2\)

### Step 2: Write the discriminant
The discriminant \(D\) of a quadratic equation is given by:

\[ D = B^2 - 4AC \]

Substituting the values of \(A\), \(B\), and \(C\):

\[ D = [-2b(a + c)]^2 - 4(a^2 + b^2)(b^2 + c^2) \]

### Step 3: Simplify the discriminant
Calculating \(D\):

\[ D = 4b^2(a + c)^2 - 4(a^2 + b^2)(b^2 + c^2) \]

### Step 4: Set the discriminant to zero
For the roots to be equal, we set the discriminant \(D\) to zero:

\[ 4b^2(a + c)^2 - 4(a^2 + b^2)(b^2 + c^2) = 0 \]

Dividing the entire equation by 4:

\[ b^2(a + c)^2 - (a^2 + b^2)(b^2 + c^2) = 0 \]

### Step 5: Expand and rearrange
Expanding the right-hand side:

\[ b^2(a + c)^2 = a^2b^2 + b^4 + b^2c^2 \]

Now, we can rewrite the equation:

\[ b^2(a^2 + 2ac + c^2) = a^2b^2 + b^4 + b^2c^2 \]

### Step 6: Cancel common terms
Subtract \(a^2b^2 + b^2c^2\) from both sides:

\[ b^2(2ac) = b^4 \]

### Step 7: Factor out \(b^2\) (assuming \(b \neq 0\))
Dividing both sides by \(b^2\):

\[ 2ac = b^2 \]

### Conclusion
Thus, we arrive at the condition:

\[ ac = \frac{b^2}{2} \]

### Final Answer
The correct condition for the roots of the equation to be equal is:

\[ ac = b^2 \]