Question

Cum se rezolva sistemul:
X+y=4
Xy=3

Answer 1

   
[tex]\displaystyle\\ \left\{ {{x+y=4} \atop {xy=3}} \right. \\ _\texttt{Este un sistem de ecuatii de gradul 2 deoarece contine produsul a doua necunoscute.}\\ _\texttt{Este un sistem simetric de ecuatii. Solutiile pot comuta intre ele}\\ _\text{deoarece adunarea si inmultirea sunt comutative.}\\\\ _\text{Vom rezolva sistemul de gradul 2, transformandu-l in ecuatie de gradul 2.} [/tex]


[tex]\displaystyle\\ \text{Rezolvare:}\\\\ \text{Scriem ecuatia de gradul 2 sub forma:}\\\\ x^2 - Sx + P=0\\\\ \text{unde S este suma radacinilor si P este produsul radacinilor.}\\ S = x_1+x_2=4\\ P = x_1x_2=3\\\\ \Longrightarrow~~x^2-4x+3=0\\\\ x_{12}= \frac{-b\pm \sqrt{b^2-4ac}}{2a}=\\\\ = \frac{4\pm \sqrt{16-12}}{2} =\frac{4\pm \sqrt{4}}{2}=\frac{4\pm 2}{2}=2\pm 1\\\\ x_1 = 2+1 = \boxed{\bf 3}\\ y_1 = 4 - 3 = \boxed{\bf 1} \\\\ x_2 = 2-1 = \boxed{\bf 1}\\ y_2 = 4 - 1 = \boxed{\bf 3} \\\\ [/tex]

Answer 2

x+y=4
xy=3
1+3=4
xy=3
1*3=3
=>x=1,y=3

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3+1=4
3*1=3
=>a=3,b=1