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Type of equations, separated by degree:

Equation of first degree, linear equation:

kx + d = 0

Equation of second degree, quadratic equation:

ax2 + bx + c = 0

Equation of 3rd degree, cubic equation:

ax3 + bx2 + cx + d = 0

When applied it gives a scheme to correctly calculate a figure in a coordinate system.


As correctly pointed out by BrianShader, the list goes on forever. After the cubic equation there is the quartic equation, then there comes the quintic...

Bernoulli's Method for the Solution of Polynomial Equations

This is elegant silly.

The solutions of the equation x2 - x - 1 = 0 are (1 + √5)/2 and (1 - √5)/2, the long side and the short side of the Golden Mean respectively. That is (1 + √5) : 1 and 1 : (1 - √5) are both the Golden Mean.

The Corresponding Difference Equation:

sn - sn-1 - sn-2 = 0 when rearanged means the next element of the sequence equals the sum of the previous two - the famous Fibonacci sequence.

The Fibonacci sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21 etc....

Form the corresponding sequence of quotients:

1, 2, 1.5, 5/3, 8/5, 13/8, 21/13 etc....

The sequence of quotients converges to one of the roots of the polynomial equation.

But every polynomial equation has an analogous difference equation. We may extract a root from any polynomial equation therefore, as long as the sequence of quotients converges.

The QD algorithm is a modern version of this which extracts ?all the roots and ?always converges (Help me Mother!)

Isn't life grand.

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