Let a,b,c , d ar positive real number such that `a/b ne c/d,` then the roots of the equation: `(a^(2) +b^(2)) x ^(2) +2x (ac+ bd) + (c^(2) +d^(2))=0` are :

Options

A. real and distinct
B. ral and equal
C. imaginary
D. nothing can be said

Answer

To solve the given quadratic equation

\[ (a^2 + b^2)x^2 + 2x(ac + bd) + (c^2 + d^2) = 0, \]

we need to analyze the discriminant of the equation to determine the nature of its roots.

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

### 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 \(C\):

\[ D = (2(ac + bd))^2 - 4(a^2 + b^2)(c^2 + d^2). \]

### Step 3: Expand the discriminant
Now, we will expand the discriminant:

\[ D = 4(ac + bd)^2 - 4(a^2 + b^2)(c^2 + d^2). \]

This simplifies to:

\[ D = 4[(ac + bd)^2 - (a^2 + b^2)(c^2 + d^2)]. \]

### Step 4: Further simplify the expression
Next, we expand \((ac + bd)^2\):

\[ (ac + bd)^2 = a^2c^2 + 2abcd + b^2d^2. \]

Now substituting this back into the discriminant:

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

### Step 5: Combine like terms
Combining the terms gives:

\[ D = 4[2abcd - a^2d^2 - b^2c^2]. \]

### Step 6: Factor out common terms
Rearranging gives:

\[ D = 4[2abcd - (ad)^2 - (bc)^2]. \]

### Step 7: Analyze the discriminant
Since \(a, b, c, d\) are positive real numbers and \( \frac{a}{b} \neq \frac{c}{d} \), we can conclude that:

\[ (ad - bc)^2 > 0. \]

Thus, \(2abcd - (ad)^2 - (bc)^2 < 0\).

### Step 8: Conclusion about the roots
Since the discriminant \(D < 0\), the roots of the quadratic equation are imaginary.

### Final Answer
The roots of the equation are **imaginary**.

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