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.
\[ \therefore { The roots are of opposite signs.} \]