Question

Roots of the equation \( x^2 + bx - c = 0 \) (\( b, c>0 \)) are:

• A. Both positive
• B. Both negative
• C. Of opposite sign
• D. None of the above

Hint:

For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots is given by: \[ \alpha + \beta = -\frac{b}{a} \] and the product of the roots is: \[ \alpha \beta = \frac{c}{a} \]

Answer 1

The Correct Option is C

Step 1: {Understand the nature of roots}
We know that if the roots of a quadratic equation are of the same sign, then the product of the roots is positive. If the roots are of opposite signs, then their product is negative.
Step 2: {Apply the formula for product of roots}
\[ \alpha \beta = \frac{-c}{1} = -c \] Since \( c>0 \), the product of roots is negative.
\[ \therefore { The roots are of opposite signs.} \]

Answer 2

Step 1: Analyze the given quadratic equation
The equation is:
\[ x^2 + bx - c = 0 \] where \( b > 0 \) and \( c > 0 \).

Step 2: Use the relationship between roots and coefficients
For a quadratic equation \( x^2 + bx + c = 0 \), the sum and product of roots are:
\[ \text{Sum of roots} = -b \] \[ \text{Product of roots} = -c \]

Step 3: Interpret the signs of sum and product
Since \( b > 0 \), sum of roots = \( -b < 0 \)
Since \( c > 0 \), product of roots = \( -c < 0 \)

Step 4: Conclusion about the roots
The product of roots is negative, which means the roots have opposite signs.

✅ Final Answer: The roots are of opposite sign.