Question

if difference of roots of the equation x²-px+q=0 is 1 then p²+4q²equals

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{roots}\mathrm{are}\mathrm{m},\mathrm{n} \\ \mathrm{m}-\mathrm{n}=1 \\ {\mathrm{x}}^2-\mathrm{px}+\mathrm{q}=0 \\ \mathrm{So},\mathrm{m}+\mathrm{n}=\mathrm{p}\mathrm{and}\mathrm{mn}=\mathrm{q} \\ \mathrm{m}-\mathrm{n}=\sqrt{{\left(\mathrm{m}+\mathrm{n}\right)}^2-4\mathrm{mn}}=\sqrt{{\mathrm{p}}^2-4\mathrm{q}}=1 \\ \mathrm{So},{\mathrm{p}}^2-4\mathrm{q}=1 \\ \mathrm{Adding}4{\mathrm{q}}^2+4\mathrm{q}\mathrm{bothside}\mathrm{we}\mathrm{get} \\ {\mathrm{p}}^2+4{\mathrm{q}}^2=1+4{\mathrm{q}}^2+4\mathrm{q}={\left(2\mathrm{q}+1\right)}^2 \end{gathered}$$