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Question

# If the zeroes of the quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the _______ sign.

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Solution

## Let f(x) = ax2 + bx + c $\mathrm{Let}\mathrm{the}\mathrm{zeroes}\mathrm{of}f\left(x\right)\mathrm{be}\alpha ,\beta ,\mathrm{where}\mathrm{all}\mathrm{these}\mathrm{zeroes}\mathrm{are}\mathrm{negative}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Then},\phantom{\rule{0ex}{0ex}}\alpha +\beta =-\frac{b}{a}\phantom{\rule{0ex}{0ex}}\because \alpha ,\beta \mathrm{are}\mathrm{negative}\phantom{\rule{0ex}{0ex}}\therefore \frac{b}{a}\mathrm{is}\mathrm{positive}\phantom{\rule{0ex}{0ex}}⇒\mathrm{either}\mathrm{both}a\mathrm{and}b\mathrm{are}\mathrm{positive}\mathrm{or}\mathrm{both}\mathrm{are}\mathrm{negative}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\alpha \beta =\frac{c}{a}\phantom{\rule{0ex}{0ex}}\because \alpha ,\beta \mathrm{are}\mathrm{negative}\phantom{\rule{0ex}{0ex}}\therefore \frac{c}{a}\mathrm{is}\mathrm{positive}\phantom{\rule{0ex}{0ex}}⇒\mathrm{either}\mathrm{both}a\mathrm{and}c\mathrm{are}\mathrm{positive}\mathrm{or}\mathrm{both}\mathrm{are}\mathrm{negative}.$ Hence, if the zeroes of the quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the same sign.

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