Question

Уравнение 2x^2+5x+1=0 имеет корни x1 и x2. Найдите значение выражения x1*x2^2+x1^2*x2.

Answer 1

2x^2+5x+1=0

Д=5^2-4*2*1=25-8=17

x1=(-5+√17)/4

x2=(-5-√17)/4

 

$<var>x_1*x_2^2+x_1^2*x_2=\frac{-5+\sqrt{17}}{4}*(\frac{-5-\sqrt{17}}{4})^2+(\frac{-5+\sqrt{17}}{4})^2*\frac{-5-\sqrt{17}}{4}=\\ =-\frac{5-\sqrt{17}}{4}*(-\frac{5+\sqrt{17}}{4})^2+(-\frac{5-\sqrt{17}}{4})^2*(-\frac{5+\sqrt{17}}{4})=\\ =-\frac{5-\sqrt{17}}{4}*(-\frac{5+\sqrt{17}}{4})*((-\frac{5+\sqrt{17}}{4})+(-\frac{5-\sqrt{17}}{4}))=\\ =\frac{(5-\sqrt{17})(5+\sqrt{17})}{16}*(-\frac{5+\sqrt{17}+5-\sqrt{17}}{4})=\\ =\frac{25-(\sqrt{17})^2}{16}*(-\frac{10}{4})=\frac{25-17}{16}*(-\frac{10}{4})=\\ </var>$

$<var>=\frac{8}{16}*(-\frac{10}{4})=\frac{1}{2}*(-\frac{5}{2})=-\frac{5}{4}\\</var>$

Answer 2

 

$<var>2x^2+5x+1=0\\\\ x_1*x_2 = \frac{1}{2}\\\\ x_1+x_2 = -\frac{5}{2}\\\\ x_1+x_2 = -\frac{5}{2} | * x_1*x_2\\\\ x_1^2x_2 + x_1x_2^2 = -\frac{5}{2}*x_1*x_2\\\\ x_1^2x_2 + x_1x_2^2 = -\frac{5}{2}*\frac{1}{2}\\\\ x_1^2x_2 + x_1x_2^2 = -\frac{5}{4} </var>$