# 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.

Enter an expression:

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## Solution

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