Question

x+y=10
x*y=21
Sunt in sistem .Sa se rezolve sistemul

Answer 1

 

$\displaystyle\\\text{Transformam sistemul de ecuatii simetrice de 2 necunoscute de gradul 2}\\\text{intr-o ecuatie de gradul 2 cu o singura necunoscuta astfel:}\\\\\text{x si y sunt solutiile.}\\\\S = x+y = 10~~~(SUMA)\\P = x \cdot y = 21~~~(PRODUSUL)\\\\\text{Scriem ecuatia de gradul 2 sub forma:}\\\\x^2-Sx+P=0,~~~\text{unde}~~S=10;~~P=21$

$\displaystyle\\x^2-10x+21=0\\\\\text{Rezolvam ecuatia de gradul 2:}\\\\x_{12}=\frac{-b\pm\sqrt{b^2-4ac}}{2}= \frac{10\pm\sqrt{100-84}}{2}=\\\\=\frac{10\pm\sqrt{16}}{2}=\frac{10\pm4}{2}=5\pm2\\x_1=5+2=7\\\\x_2=5-2=3\\\text{Solutiile sistemului:}\\\\S1:\\x=7\\y=3\\\\S2:\\x=3\\y=7\\\\\text{Solutiile comuta deoarece sistemul este simetric}$

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