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Student[Basics]

 CompleteSquareSteps
 generate steps for completing the square of a quadratic expression

 Calling Sequence CompleteSquareSteps( expr, variable ) CompleteSquareSteps( expr, implicitmultiply = true )

Parameters

 expr - string or expression variable - (optional) variable to collect the terms by implicitmultiply - (optional) truefalse output = ... - (optional) option to control the return value displaystyle = ... - (optional) option to control the layout of the steps

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right):$
 > $\mathrm{CompleteSquareSteps}\left(\left(x+1\right)\left(x-3\right)=4x+2\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]{=}\left[{}\right]\\ \text{•}& {}& \text{Bring terms to left-side}\\ {}& {}& \left({x}{+}{1}\right){}\left({x}{-}{3}\right){-}{4}{}{x}{-}{2}{=}{0}\\ \text{•}& {}& \text{Collect in terms of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\\ {}& {}& {{x}}^{{2}}{-}{6}{}{x}{-}{5}{=}{0}\\ \text{•}& {}& \text{Add and subtract}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\left(\frac{-6}{2}\right)}^{2}\\ {}& {}& \left[{}\right]{=}{0}\\ \text{•}& {}& \text{Simplify terms}\\ {}& {}& \left[{}\right]{=}{0}\\ \text{•}& {}& \text{The first}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}3\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{terms can be regrouped as a perfect square}\\ {}& {}& \left[{}\right]{=}{0}\\ \text{•}& {}& \text{Simplify the remaining term}\\ {}& {}& \left[{}\right]{=}{0}\end{array}$ (1)
 > $\mathrm{Student}:-\mathrm{Precalculus}:-\mathrm{CompleteSquare}\left({\mathrm{sin}\left(t\right)}^{2}+2\mathrm{sin}\left(t\right)+1,\mathrm{sin}\left(t\right)\right)$
 ${\left({\mathrm{sin}}{}\left({t}\right){+}{1}\right)}^{{2}}$ (2)

Compatibility

 • The Student:-Basics:-CompleteSquareSteps command was introduced in Maple 2023.
 • For more information on Maple 2023 changes, see Updates in Maple 2023.