# Questions tagged [algebraic-manipulation]

The art of manipulating an algebraic expression into the desired form.

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### Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)

Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$. When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
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### How do I get my equation to have the form $(x-a)^2 + (y-b)^2 + (z-c)^2-d = 0$?

I want Mathematica to express the equation $$-11 - 2 x + x^2 - 4 y + y^2 - 6 z + z^2=0$$ in the form $$(x - 1)^2 + (y - 2)^2 + (z - 3)^2 - 25=0$$ How do I tell Mathematica to do that?
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### What is the proper way to verify that two expressions are equal?

First of all, it appears to me that $$\left|\frac{1}{x} + x^2\right| = \frac{|1 + x^3|}{|x|}$$ is true. I don't know why, but my attempt to verify it in Mathematica failed. ...
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### Replacing composite variables by a single variable

To replace a single variable by another variable, one can simply use the the replace all (/.) operator (e.g., ...
905 views

### Transform Root objects into Trigonometric expressions

Consider the Root objects roots = Table[Root[-1 + 27 #1^2 - 162 #1^4 + 243 #1^6 &, i],{i,1,6}] These can be expressed in terms trigonometric functions as ...
7k views

### Defining a function that completes the square given a quadratic polynomial expression

How can I write a function that would complete the square in a quadratic polynomial expression such that, for example, CompleteTheSquare[5 x^2 + 27 x - 5, x] ...
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### Square both sides of an equation?

Can I define an equation (for example, x+1 == y^2 + 2), and tell Mathematica to square both sides? If not, what is an equivalent way to achieve this?
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### Why doesn't Mathematica expand Cos[x]^3 Sin[x]^2?

I found some examples of Mathematica's commands usage in an old manual but the program gives me different result than expected ...
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### InverseSeries of multiple variables and multiple equations

CONTEXT Let us consider a bit of the Universe in which we draw spheres (see a high resolution image here). Astronomers have shown that the density within these spheres could be predicted quite ...
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### Why is ToRadicals[] not able to handle all cases? Is there a workaround?

The documentation for the function ToRadicals says: ♦ There are some cases in which expressions involving radicals can in principle be given, but ...
924 views

### Distances between points in periodic cube

Question How can one implement more efficiently/elegantly/memory savvily the following function which returns a matrix of all Euclidian distances between points in 3D within a cube of width ...
726 views

### Mathematica function equivalent to MATLAB's residue function (partial fraction expansion)

I am looking for a Mathematica function equivalent to MATLAB's residue function. If there is no Mathematica equivalent, I would like to write a function that given ...
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### How do I expand a sum?

I have a problem with Mathematica's symbolic manipulations. As an example, consider the following expression: $$\sum _{i=1}^n -2 x_i \left(-a x_i-b+y_i\right)=0$$ How do I get Mathematica to expand ...
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### Shaping/simplifying equations in a certain way

A problem I am occasionally facing is to simplify an equation not to it's shortest form but to a form that is simple by other means. Often, this is grouping the term according to certain functions, ...
874 views

### Simplify radical expression (a + b - Sqrt[a^2 - b^2])/Sqrt[a + b]

I have the following expansions: $\frac{a+b-\sqrt{a^2-b^2}}{\sqrt{a+b}}$ and $a>b>0$ Obviously we can simplify it further by hand $\sqrt{a+b}-\sqrt{a-b}$, but Mathematica doesn't know how to ...
529 views