The roots of `a^(2)x^(2)+abx+b^(2),ane0` are :

Options

A. equal
B. non - real
C. unequal
D. none of these

Answer

To find the roots of the quadratic equation \( a^2 x^2 + ab x + b^2 = 0 \), 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:
- \( A = a^2 \)
- \( B = ab \)
- \( C = b^2 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = (ab)^2 - 4(a^2)(b^2) \]

### Step 3: Simplify the discriminant
Now, we simplify the expression for the discriminant:
\[ D = a^2b^2 - 4a^2b^2 \]
This simplifies to:
\[ D = -3a^2b^2 \]

### Step 4: Analyze the discriminant
Since \( a \neq 0 \) and \( b \) can be any real number, \( -3a^2b^2 < 0 \). This means the discriminant is negative.

### Step 5: Determine the nature of the roots
When the discriminant is negative, the roots of the quadratic equation are non-real (complex). Therefore, the roots of the equation \( a^2 x^2 + ab x + b^2 = 0 \) are non-real.

### Final Answer
The roots of the equation \( a^2 x^2 + ab x + b^2 = 0 \) are non-real.

---