Question

Stiind ca x₁ si x₂ sunt solutiile x²-2014x+1=0 , sa se calculeze: $\frac{1}{x1}+ \frac{1}{x2}$

Answer 1

$Avem:~ x_{1}= \frac{-b+ \sqrt{ \Delta}}{2a} ~si~ x_{2} = \frac{-b- \sqrt{\Delta} }{2a}. \\ \\ \frac{1}{ x_{1} } + \frac{1}{ x_{2} } = \frac{ x_{1}+ x_{2} }{ x_{1}* x_{2} }= \frac{ \frac{-b+ \sqrt{\Delta} }{2a} + \frac{-b- \sqrt{\Delta} }{2a} }{ \frac{-b+ \sqrt{\Delta} }{2a}* \frac{-b-\sqrt{\Delta}}{2a} } = \frac{ \frac{-2b}{2a} }{ \frac{ b^{2}-\Delta }{4 a^{2} } } = -\frac{b}{a} * \frac{4 a^{2} }{ b^{2}-\Delta } = \\ =-\frac{b}{a} * \frac{4 a^{2} }{ b^{2}-( b^{2}-4ac) }= -\frac{4ab}{4ac}=$$\frac{b}{c} = \frac{-2014}{1}=\boxed{-2014}.$