Question

Не розв’язуючи рівняння: а). –5х² +10х –15 =0;
б). 2х² –3х +11 =0 знайдіть значення виразу 1/х1+ 1/х2та х1х2² + х2х1

Answer 1

$\displaystyle\bf\\1)\\\\-5x^{2} +10x-15=0 \ |:(-5)\\\\x^{2} -2x+3=0\\\\Teorema \ Vieta \ :\\x_{1} + x_{2} =2\\\\x_{1} \cdot x_{2} =3\\\\\\\frac{1}{x_{1} } +\frac{1}{x_{2} } =\frac{x_{1} +x_{2} }{x_{1}\cdot x_{2} } =\frac{2}{3} \\\\\\x_{1} x_{2} ^{2} +x_{2} x_{1}^{2} =x_{1} \cdot x_{2} \cdot(x_{1} +x_{2} )=3\cdot 2=6$

$\displaystyle\bf\\2)\\\\2x^{2} -3x-+11=0 \ |:2\\\\x^{2} -1,5x+5,5=0\\\\Teorema \ Vieta \ :\\x_{1} + x_{2} =1,5\\\\x_{1} \cdot x_{2} =5,5\\\\\\\frac{1}{x_{1} } +\frac{1}{x_{2} } =\frac{x_{1} +x_{2} }{x_{1}\cdot x_{2} } =\frac{1,5}{5,5} =\frac{3}{11} \\\\\\x_{1} x_{2} ^{2} +x_{2} x_{1}^{2} =x_{1} \cdot x_{2} \cdot(x_{1} +x_{2} )=5,5\cdot 1,5=8,25$