Completing the Square Calculator
The calculator will try to complete the square for the given quadratic expression, ellipse, hyperbola, or any polynomial expression, with steps shown.
Solution
Your input: complete the square in $$$x^{2} - 4 x + 5 y^{2} + 10 y + 14$$$.
Add and subtract $$$5$$$:
$$$x^{2} - 4 x + 5 y^{2} + 10 y + 14=x^{2} - 4 x + 5 y^{2} + 10 y + 14 + \color{red}{\left(5\right)} - \color{red}{\left(5\right)}$$$
Factor $$$5$$$:
$$$x^{2} - 4 x + 9 + \color{red}{\left(5 y^{2} + 10 y + 5\right)}=x^{2} - 4 x + 9 + \color{red}{\left(5 \left(y^{2} + 2 y + 1\right)\right)}$$$
Complete the square:
$$$x^{2} - 4 x + 9 + 5 \color{red}{\left(y^{2} + 2 y + 1\right)}=x^{2} - 4 x + 9 + 5 \color{red}{\left(y + 1\right)^{2}}$$$
Add and subtract $$$4$$$:
$$$x^{2} - 4 x + 5 \left(y + 1\right)^{2} + 9=x^{2} - 4 x + 5 \left(y + 1\right)^{2} + 9 + \color{red}{\left(4\right)} - \color{red}{\left(4\right)}$$$
Complete the square:
$$$5 \left(y + 1\right)^{2} + 5 + \color{red}{\left(x^{2} - 4 x + 4\right)}=5 \left(y + 1\right)^{2} + 5 + \color{red}{\left(x - 2\right)^{2}}$$$
Answer: $$$x^{2} - 4 x + 5 y^{2} + 10 y + 14=\left(x - 2\right)^{2} + 5 \left(y + 1\right)^{2} + 5$$$.