The roots of the equation `2a^(2) x^(2) - 2abx + b^(2) = 0`. When `a lt 0 and b gt 0` are

Options

A. a) Sometimes complex
B. b) Always complex
C. c) Always irrational
D. d) Always real

Answer

To find the roots of the quadratic equation \(2a^2 x^2 - 2abx + b^2 = 0\) given that \(a < 0\) and \(b > 0\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be compared to the standard form \(Ax^2 + Bx + C = 0\), where:
- \(A = 2a^2\)
- \(B = -2ab\)
- \(C = b^2\)

### 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 \(C\):
\[ D = (-2ab)^2 - 4(2a^2)(b^2) \]
Calculating \(D\):
\[ D = 4a^2b^2 - 8a^2b^2 \]
\[ D = -4a^2b^2 \]

### Step 3: Analyze the discriminant
Since \(a < 0\), \(a^2\) is positive. Also, since \(b > 0\), \(b^2\) is positive. Therefore, the product \(a^2b^2\) is positive, and thus:
\[ D = -4a^2b^2 < 0 \]
This indicates that the discriminant is negative.

### Step 4: Conclusion about the roots
When the discriminant \(D < 0\), it implies that the quadratic equation has complex roots. Therefore, the roots of the equation \(2a^2 x^2 - 2abx + b^2 = 0\) are complex.

### Final Answer
The roots of the equation are complex.

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