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