Question

suma patratelor solutiilor ecuatiei x la a 2 -2x+m =0 este egala cu 16. Sa se afla m

Answer 1

x^2-2x+m=0

x1+x2= -b/a

x1+x2= 2
x1^2+x2^2=16

x1=2-x2
(2-x2)^2+x^2=16
4-4x2+x2^2=16
2x2^2-4x2-12=0 / :2
x2^2-2x2=6 / +1
x2^2-2x2+1=7
(x2-1)^2=7

x1= 1+√7
x2= 1-√7 

x^2-2x+m=0
x= 1+√7 => (1+√7)^2-2(1+√7)+m=0 => m = -6

Answer 2

x²-2x+m=0, avem suma radacinilor S=$x_{1} + x_{2}=- \frac{b}{a}=2,produsul,P= x_{1} x_{13}= \frac{c}{a}=m$
$x^2_{1} + x^2_{2}=( x_{1}+ x_{2})^2-2 x_{1} x_{2}=S^2-2P=4-2m=16$,  deci 2m=-12 adica m=-6.