question_answer

Find a quadratic polynomial, the sum and product of whose zeroes are 0 and \[-\frac{3}{5}\] respectively. Hence find the zeroes.

Answer:

Quadratic polynomial

\[={{x}^{2}}-\] (Sum of zeroes) x + Product of zeroes

\[={{x}^{2}}-(0)x+\left( \frac{-3}{5} \right)={{x}^{2}}-\frac{3}{5}\]

\[={{(x)}^{2}}-\left( \sqrt{\frac{3}{5}} \right)\]

=\[\left( x-\sqrt{\frac{3}{5}} \right)\left( x+\sqrt{\frac{3}{5}} \right)\left[ \begin{align}   & \text{By}\,\,\text{applying} \\  & \left( {{a}^{2}}-{{b}^{2}} \right)=(a+b)(a-b) \\ \end{align} \right]\]

Zeroes are, \[x-\sqrt{\frac{3}{5}}=0\] or \[x+\sqrt{\frac{3}{5}}=0\]

\[\Rightarrow x=\sqrt{\frac{3}{5}}\] or \[\Rightarrow x=-\sqrt{\frac{3}{5}}\]

\[x=\sqrt{\frac{3}{5}\times \frac{5}{3}}\] or \[x=-\sqrt{\frac{3}{5}\times \frac{5}{3}}\]

\[x=\frac{\sqrt{15}}{5}\] or \[x=\frac{-\sqrt{15}}{5}\]