Question

Rezolvati in R ecuatia √(x²-3)-x=1

Answer 1

 √(x²-3)-x=1
√x²-3=1+x
x²-3=1+2x+x²
-3=1+2x
-2x=1+3
x=-2

dar
√(-2)²-3-(-2)=1
3=1
x≠2
x ∈  Ø

Answer 2

$\sqrt{x^2-3}-x=1 \\ \\ \sqrt{x^2-3} = x+1 \\ \\ Conditii de existenta: \left\{ \begin{array}{c} x^2-3 \ \textgreater \ 0 \\ x+1 \ \textgreater \ 0 \end{array} \right \Rightarrow \left\{ \begin{array}{c} x^2 \ \textgreater \ 3 \\ x \ \textgreater \ -1 \end{array} \right \Rightarrow \\ \\ \Rightarrow \left\{ \begin{array}{c} x\in (-\infty,\sqrt3)\cup(\sqrt3,+\infty) \\ x\in (-1,+\infty) \end{array} \right| \Rightarrow D = (\sqrt3,+\infty)$

$\sqrt{x^2-3} = x+1\Big|^2 \\ \\ x^2-3 = x^2+2x+1 \\ \\ x^2-x^2-2x = 1+3 \\ \\ -2x = 4 \\ \\ x = \dfrac{4}{-2} \\ \\ x = -2 \notin D \\ \\ \boxed{S = \emptyset}$