Vervollständigung des Quadratrechners

Your input: complete the square in $x^{2} - 4 x + 5 y^{2} + 10 y + 14$.

Add and subtract $4$:

$x^{2} - 4 x + 5 y^{2} + 10 y + 14=x^{2} - 4 x + 5 y^{2} + 10 y + 14 + {\color{red}{\left(4\right)}} - {\color{red}{\left(4\right)}}$

Complete the square:

$5 y^{2} + 10 y + 10 + {\color{red}{\left(x^{2} - 4 x + 4\right)}}=5 y^{2} + 10 y + 10 + {\color{red}{\left(x - 2\right)^{2}}}$

Add and subtract $5$:

$5 y^{2} + 10 y + \left(x - 2\right)^{2} + 10=5 y^{2} + 10 y + \left(x - 2\right)^{2} + 10 + {\color{red}{\left(5\right)}} - {\color{red}{\left(5\right)}}$

Factor $5$:

$\left(x - 2\right)^{2} + 5 + {\color{red}{\left(5 y^{2} + 10 y + 5\right)}}=\left(x - 2\right)^{2} + 5 + {\color{red}{\left(5 \left(y^{2} + 2 y + 1\right)\right)}}$

Complete the square:

$\left(x - 2\right)^{2} + 5 + 5 {\color{red}{\left(y^{2} + 2 y + 1\right)}}=\left(x - 2\right)^{2} + 5 + 5 {\color{red}{\left(y + 1\right)^{2}}}$

Answer: $x^{2} - 4 x + 5 y^{2} + 10 y + 14=\left(x - 2\right)^{2} + 5 \left(y + 1\right)^{2} + 5$.