The CompleteSquareSteps command accepts a polynomial and displays the steps required to complete the square.  As a pre-step, the given expression will be reorganized into the general form of a quadratic by expanding and simplifying as needed.  

An optional variable can be provided as a second argument.  This so called variable can also be an expression, such as sin(t) for completing the square to the form (sin(t)+a)^2+b. If no variable is provided, the return of indets(expression,name) will be used to find the first variable that has has degree 2.

If expr is a string, then it is parsed into an expression using InertForm:-Parse so that no automatic simplifications are applied, and thus no steps are missed.  

The implicitmultiply option is only relevant when expr is a string.  This option is passed directly on to the InertForm:-Parse command and will cause things like 2x to be interpreted as 2*x, but also, xyz to be interpreted as x*y*z.

The output and displaystyle options are described in Student:-Basics:-OutputStepsRecord. The return value is controlled by the output option.  

This function is part of the Student:-Basics package.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right)\:$

$\mathrm{CompleteSquareSteps}\left(\left(x\+1\right)\left(x-3\right)\=4x\+2\right)$

$\begin{array}{lll}& & \left[\right]=\left[\right]\\ \text{•}& & \text{Bring terms to left-side}\\ & & \left(x+1\right)\left(x-3\right)-4x-2=0\\ \text{•}& & \text{Collect in terms of}x\\ & & x^2-6x-5=0\\ \text{•}& & \text{Add and subtract}{\left(\frac{\mathrm{-6}}{2}\right)}^2\\ & & \left[\right]=0\\ \text{•}& & \text{Simplify terms}\\ & & \left[\right]=0\\ \text{•}& & \text{The first}3\text{terms can be regrouped as a perfect square}\\ & & \left[\right]=0\\ \text{•}& & \text{Simplify the remaining term}\\ & & \left[\right]=0\end{array}$

$\mathrm{Student}:-\mathrm{Precalculus}:-\mathrm{CompleteSquare}\left({\sin \left(t\right)}^2\+2\sin \left(t\right)\+1\,\sin \left(t\right)\right)$

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

The Student:-Basics:-CompleteSquareSteps command was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.