The CompleteSquare(f, x) command, where f is a quadratic expression, equation, or an inequality in x, completes the square of f.  If f is not itself quadratic in x, its subexpressions are examined, and any which are quadratic in x are transformed to completed square form.

If x is a list or set, the operation of completing the square is applied successively to each component of x.  These components can be names, such as $u$ and $v$, or functions, such as $\sin \left(t\right)$ and $\frac{\ⅆ}{\ⅆw}h\left(w\right)$.

The CompleteSquare(f) command is equivalent to CompleteSquare(f, indets(f,name)).

The command CompleteSquare(f, [x, y]) is equivalent to the command CompleteSquare(f, x, y).

The completed square form of the general quadratic expression $a{x}^2+bx+c$ is $a{\left(x+\frac{b}{2a}\right)}^2+c-\frac{b^2}{4a}$.

For a step-by-step solution that completes the square, see CompleteSquareSteps.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Precalculus}\right]\right)\:$

$\mathrm{CompleteSquare}\left(3x^2+2x\,x\right)$

$3{\left(x+\frac{1}{3}\right)}^2-\frac{1}{3}$

$\mathrm{CompleteSquare}\left(x^2{\left(1-y\right)}^3+2y^{2}x-y\,x\right)$

${\left(1-y\right)}^3{\left(x+\frac{y^2}{{\left(1-y\right)}^3}\right)}^2-y-\frac{y^4}{{\left(1-y\right)}^3}$

$\mathrm{CompleteSquare}\left(\frac{1}{{\sin \left(t\right)}^2+2\sin \left(t\right)+1}\,\sin \left(t\right)\right)$

$\frac{1}{{\left(\sin \left(t\right)+1\right)}^2}$

$\mathrm{CompleteSquare}\left(a^2+b^2+a+b\,a\,b\right)$

${\left(b+\frac{1}{2}\right)}^2+{\left(a+\frac{1}{2}\right)}^2-\frac{1}{2}$

$\mathrm{CompleteSquare}\left(x^2-x=y^2+2y+3\right)$

${\left(x-\frac{1}{2}\right)}^2-\frac{1}{4}={\left(y+1\right)}^2+2$

$\mathrm{CompleteSquare}\left(\mathrm{Int}\left(\mathrm{Int}\left(\frac{u^2+2u}{v^2-3v}\,v\right)\,u\right)\,\left[u\,v\right]\right)$

$\int \int \left(\frac{{\left(u+1\right)}^2}{{\left(v-\frac{3}{2}\right)}^2-\frac{9}{4}}-\frac{1}{{\left(v-\frac{3}{2}\right)}^2-\frac{9}{4}}\right)\ⅆv\ⅆu$