Question

if the two equation x2-cx+d=0 x2 -ax + b = 0have one common root and the second has equal roots, then 2(b+d)= ?

Answer 1

$$\begin{gathered} Lettherootsof{x}^2-cx+d=0be\alpha and\beta \\ Andthatof{x}^2-ax+b=0be\beta and\beta \\ Nowrootsof{x}^2-ax+b=0is\beta and\beta \\ \therefore \beta +\beta =a\Rightarrow 2\beta =a\Rightarrow \beta =\frac{a}{2} \\ And\beta \times \beta =b\Rightarrow \frac{a}{2}\times \frac{a}{2}=b\Rightarrow b=\frac{a^2}{4}..........\left(1\right) \\ Now\sin ce\beta isalsoarootofthefirstequation \\ Hence\beta =\frac{a}{2}mustsatisfytheequation \\ Nowputting\beta =\frac{a}{2}in{x}^2-cx+d=0weget \\ \frac{a^2}{4}-\frac{a}{2}\times c+d=0 \\ \Rightarrow d=\frac{ac}{2}-\frac{a^2}{4}........\left(2\right) \\ Adding\left(1\right)and\left(2\right)weget \\ b+d=\frac{ac}{2} \\ \Rightarrow 2(b+d)=ac \end{gathered}$$