TeX Quellcode:

\begin{array}{rclll} \sqrt{x}\sqrt{x+1}-1 &=& -\sqrt{x}\sqrt{x+2} &\vert & +1+\sqrt{x}\sqrt{x+2}\\\\\sqrt{x}\sqrt{x+1}+\sqrt{x}\sqrt{x+2} &=& 1 \\\\\sqrt{x}\left(\sqrt{x+1}+\sqrt{x+2}\right) &=& 1 &\vert & ()^2\\\\x\left(x+1+2\sqrt{x+1}\sqrt{x+2}+x+2\right) &=& 1 \\\\2x^2+3x+2x\sqrt{x+1}\sqrt{x+2} &=& 1 &\vert & -2x^2-3x\\\\2x\sqrt{x+1}\sqrt{x+2} &=& -2x^2-3x+1 &\vert & ()^2\\\\4x^2\left(x+1\right)\left(x+2\right) &=& \left(-2x^2-3x+1\right)\left(-2x^2-3x+1\right)\\\\4x^2\left(x^2+3x+2\right) &=& 4x^4+6x^3-2x^2+6x^3+9x^2-3x-2x^2-3x+1 \\\\4x^4+12x^3+8x^2 &=& 4x^4+12x^3+5x^2-6x+1 &\vert& -4x^4-12x^3-5x^2+6x-1\\\\3x^2+6x-1 &=& 0 &\vert & :3\\x^2+2x-\dfrac{1}{3} &=& 0 &\vert & \text{p-q-Formel}\\x_{1,2} &=& -1\pm\sqrt{1+\dfrac{1}{3}}\\\\x_{1,2} &=& -1\pm\sqrt{\dfrac{4}{3}} \\\\ x_1 &=& -1+\sqrt{\dfrac{4}{3}} = -1+\dfrac{2\sqrt{3}}{3} \;\in\;\mathbb{D}\\x_2 &=& -1-\sqrt{\dfrac{4}{3}} = -1-\dfrac{2\sqrt{3}}{3} \;\not\in\;\mathbb{D}\end{array}