Question

Find a quadratic polynomial, the sum and product of whose zeroes are respectively:

(i) 5 and $-$ 50
(ii) $-$ 11 and 10

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right) \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{a}\mathrm{quadratic}\mathrm{polynomial}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}{x}^2-\left(\alpha +\beta \right)x+\alpha \beta . \\ \mathrm{Here},\alpha +\beta \mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{zeroes}\mathrm{and}\alpha \beta \mathrm{is}\mathrm{the}\mathrm{product}\mathrm{of}\mathrm{zeroes}. \\ \mathrm{Given}:\alpha +\beta =5\mathrm{and}\alpha \beta =-50 \\ \therefore \mathrm{Required}\mathrm{quadratic}\mathrm{polynomial}=x^2-\left(5\right)x+\left(-50\right)=x^2-5x-50 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{a}\mathrm{quadratic}\mathrm{polynomial}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}{x}^2-\left(\alpha +\beta \right)x+\alpha \beta . \\ \mathrm{Here},\alpha +\beta \mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{zeroes}\mathrm{and}\alpha \beta \mathrm{is}\mathrm{the}\mathrm{product}\mathrm{of}\mathrm{zeroes}. \\ \mathrm{Given}:\hspace{0.17em}\alpha +\beta =-11\mathrm{and}\alpha \beta =10 \\ \therefore \mathrm{Required}\mathrm{quadratic}\mathrm{polynomial}=x^2-\left(-11\right)x+10=x^2+11x+10 \end{gathered}$$