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mercury (planet)

In abstract algebra, the Quaternionic roots of polynomials are the roots of a polynomial in the quaternions, which are the quaternions that would make a polynomial equal to zero when substituted in place of the variable.

Quadratic function
For a quadratic polynomial $$ax^2 + bx +c$$, the roots can be found by substituting a quaternion $$t + ui + vj +wk$$ into the variable, expanding it, and then separating the real and imaginary parts and setting all of them equal to zero in a system of equations.
 * $$a(t + ui + vj +wk)^2 + b(t + ui + vj +wk) + c = 0$$
 * $$a(t^2 - u^2 - v^2 - w^2 + 2tui + 2tvj + 2twk) + b(t + ui + vj +wk) + c = 0$$
 * $$at^2 - au^2 - av^2 - aw^2 + 2atui + 2atvj + 2atwk + bt + bui + bvj +bwk + c = 0$$

\begin{cases} & at^2 - au^2 - av^2 - aw^2 +bt + c = 0 \\ & 2atui +bui = 0 \\ & 2atvj +bvj = 0 \\ & 2atwk +bwk = 0 \end{cases} $$ Taking one of the last three equations and cancelling out the common factor shows that $$t = -\frac{b}{2a}.$$ This value obtained for $$t$$ can be substituted into the first equation, which is then simplified.
 * $$a \left(-\frac{b}{2a}\right)^2 - au^2 - av^2 - aw^2 +b \left(-\frac{b}{2a}\right) + c = 0 \quad \Rightarrow \quad u^2 + v^2 + w^2 = \frac{4ac - b^2}{4a^2} $$

This shows that the roots of a quadratic $$ax^2 + bx +c$$ will be in the form $$t + ui + vj +wk$$ where $$t = -\frac{b}{2a}$$ and $$u^2 + v^2 + w^2 = \frac{4ac - b^2}{4a^2}.$$ This applies to quadratics with negative discriminants. For quadratics with positive discriminants, the hyperbolic quaternions can be used and this time the sum of the squares of the imaginary parts will be equal to $$\frac{b^2 - 4ac}{4a^2}.$$

Other degrees
This can be generalized to higher degree polynomials since all polynomials can be factored into quadratics, with an additional linear factor if the degree is odd.

Pages to create

 * Quaternionic roots of polynomials
 * Inverse Pythagorean theorem
 * Illogicopedia
 * Pierce expansion
 * Cotangent expansion
 * Orders of magnitude (angle)
 * Orders of magnitude (surface tension)
 * Orders of magnitude (current density)
 * Orders of magnitude (linear charge density)
 * Orders of magnitude (area charge density)
 * Orders of magnitude (volume charge density)
 * Capitol Green Apartments

= List of derived Planck units =

Derived Planck units are units of measurement derived from the five base Planck units. They can be expressed as a product of one or more of the base units, possibly scaled by an appropriate power of exponentiation.

Kinematic units
The following table lists various kinematic derived Planck units.

Mechanical units
The following table lists various mechanical derived Planck units.

Electromagnetic units
The following table lists various electromagnetic derived Planck units.

Thermodynamic units
The following table lists various thermodynamic derived Planck units.

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