Question

sa se rezolve ecuatiile:√x+1+√x+3=2

Answer 1

$\sqrt{x+1}+\sqrt{x+3}=2 \Big|^2 \\ \\ \Big(\sqrt{x+1}\Big)^2+2\cdot \sqrt{x+1}\cdot \sqrt{x+3}+\Big(\sqrt{x+3}\Big)^2 = 4 \\ \\ x+1+2\sqrt{(x+1)(x+3)}+x+3 = 4 \\ \\ 2x+4+2\sqrt{(x+1)(x+3)} = 4 \\ \\ 2\sqrt{(x+1)(x+3)} = -2x\Big|:2 \\ \\ \sqrt{(x+1)(x+3)} = -x,\quad \Rightarrow -x\geq 0 \Rightarrow x \leq 0 \\ \\ \sqrt{(x+1)(x+3)} = -x\Big|^2 \\ \\ \Big(\sqrt{(x+1)(x+3)}\Big)^2 = (-x)^2 \\ \\ (x+1)(x+3) = x^2 \\ \\ x^2+3x+x+3 = x^2 \\ \\ 4x+3= 0 \\ \\ 4x = -3 \\ \\ \boxed{x = -\dfrac{3}{4}}$