# 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.

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