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: {Characterize roots based on sign}
Quadratic equation roots share the same sign if their product is positive. Conversely, roots have opposite signs if their product is negative.
Step 2: {Utilize product of roots formula}
\[\alpha \beta = \frac{-c}{1} = -c\]Given that \( c>0 \), the product of roots is negative.
\[\therefore { The roots possess dissimilar signs.}\]