If the roots of the equation
`(a^(2)+b^(2)) x^(2)-2(bc+ad)x+(c^(2)+d^(2))=0` be real, then they will be equal as well and then `(a)/(b)=(d)/(c )`

Answer

To solve the given problem, we need to analyze the quadratic equation:

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

We want to determine the conditions under which the roots of this equation are real and equal.

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

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

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

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

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

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

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

Factoring out the common term \(4\):

\[ D = 4\left[(bc + ad)^2 - (a^2 + b^2)(c^2 + d^2)\right] \]

### Step 4: Set the discriminant to zero for equal roots
For the roots to be real and equal, the discriminant must be zero:

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

### Step 5: Rearranging the equation
This implies:

\[ (bc + ad)^2 = (a^2 + b^2)(c^2 + d^2) \]

### Step 6: Apply the identity
Using the identity for the square of a sum, we can expand the left-hand side:

\[ b^2c^2 + 2abcd + a^2d^2 = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \]

### Step 7: Cancel common terms
Cancelling \(b^2c^2\) and \(a^2d^2\) from both sides gives:

\[ 2abcd = a^2c^2 + b^2d^2 \]

### Step 8: Rearranging to find the ratio
Rearranging gives us:

\[ a^2c^2 + b^2d^2 - 2abcd = 0 \]

This can be factored as:

\[ (a - b)(c - d) = 0 \]

### Step 9: Conclude the ratio
Thus, we conclude that either \(a = b\) or \(c = d\). This leads us to the final relationship:

\[ \frac{a}{b} = \frac{d}{c} \]

### Final Result
Therefore, if the roots of the given quadratic equation are real and equal, then it must hold that:

\[ \frac{a}{b} = \frac{d}{c} \]