Question

rezolvati ecuatia rad x-1=x-3

Answer 1

in urma ridicarii la patrat a expresiei obtii doua posibilitati:
1)x-1=x^2-6x+9=>x^2-7x+10=0
delta=b^2-4ac=49-40=9
x1,2=(-b+-radical din delta)/2a=(7+-3)/2=>x apartine {5;2}
2)x-1=-(x^2-6x+9)=>x^2-5x+8=0
delta=25-32=-7<0=>fara solutii
in concluzie, x apartine multimii formate din elementele 5 si 2

Answer 2

$\sqrt{x-1}=x-3 \\ \\ Domeniul de de\f finitie: x-1 \geq0 \quad si \quad x-3 \geq 0 \\ \Rightarrow x \geq 1 \quad si \quad x\geq 3 \Rightarrow D = [3, \infty) \\ \\ \sqrt{x-1}=x-3 \Big|^2 \Rightarrow \sqrt{x-1}^2 =(x-3)^2 \Rightarrow \\ \Rightarrow x-1=x^2-6x+9 \Rightarrow x^2 -7x+10 = 0 \\ \\ \Delta = 49-40 = 9 \Rightarrow x_{1,2} = \dfrac{7\pm 3}{2} \Rightarrow x_1 = 5,\quad x_2 = 2 \notin D \\ \\ \Rightarrow S = \big\{5\big\}$