If a, b `in` R, then the equation `x^(2) - abx - a^(2) = 0` has

Options

A. one positive and one negative root
B. both positive roots
C. both negative roots
D. non-real roots

Explanation

To determine the nature of the roots of the quadratic equation \( x^2 - abx - a^2 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = -ab \)
- \( c = -a^2 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-ab)^2 - 4(1)(-a^2) = a^2b^2 + 4a^2 \]

### Step 3: Factor the discriminant
We can factor out \( a^2 \) from the discriminant:
\[ D = a^2(b^2 + 4) \]

### Step 4: Analyze the discriminant
Since \( a^2 \) is always non-negative (as \( a \in \mathbb{R} \)), the sign of \( D \) depends on \( b^2 + 4 \):
- \( b^2 + 4 \) is always positive because \( b^2 \geq 0 \) and \( 4 > 0 \).
- Therefore, \( D > 0 \).

### Step 5: Conclusion about the roots
Since the discriminant \( D > 0 \), the quadratic equation has two distinct real roots.

### Step 6: Determine the nature of the roots
To find the nature of the roots, we can use Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = ab \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = -a^2 \)

Since \( -a^2 < 0 \), this implies that one root is positive and the other root is negative.

### Final Answer
Thus, the equation \( x^2 - abx - a^2 = 0 \) has two distinct real roots, one positive and one negative.

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