Completing the Square Calculator

Complete squares step by step

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