Important: The student package has been deprecated. Use the superseding command Student[Precalculus][CompleteSquare] instead.

This function completes the square of polynomials of degree 2 in x by re-writing such polynomials as perfect squares plus a remainder. If more than one variable appears in f, then x must be specified.  x can be a name, list or set.

The polynomial may appear as a subexpression of f, so that equations, reciprocals, and in general, parts of expressions, may be simplified. Since the polynomial may occur as an argument to Int this function can be used to help simplify unevaluated integrals.

Terms not involving the indicated variable are treated as constants.

The command with(student,completesquare) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{student}\right)\:$

$\mathrm{completesquare}\left(9x^2+24x+16\right)$

$9{\left(x+\frac{4}{3}\right)}^2$

$\mathrm{completesquare}\left(x^2-2xa+a^2+y^2-2yb+b^2=23\,x\right)$

${\left(x-a\right)}^2+b^2-2yb+y^2=23$

$\mathrm{completesquare}\left(\,y\right)$

${\left(y-b\right)}^2+{\left(x-a\right)}^2=23$