Solving Cubic Equations... (1 Viewer)

[hyparzero]

It is highly unlike that we need to utilise the General Solution of Cubic equations anytime soon, as its completely out of the syllabus, and creates more working compared to other methods.

however, there is a interesting method here which solves cubic equations if coefficient before x² is zero.

Let

x³ + 3ax = 2b

be the general cubic equation.

Let x = a/y - y

Hence equation becomes:

y⁶ + 2by³ - a³ = 0

which is a quadratic equation in y³; solve for y³, extract the cube root y; and compute the x- values

Can this method be used legally for ext2.

 

[buchanan]

The history of this is an important thing to consider as far as the origins of complex numbers are concerned. See the references below

There are some Newton's method approximation questions on cubics and quartics in past HSC papers, and although it's not required in these questions, nevertheless it is possible to solve them exactly using the cubic and quartic formulae:

Solve for xεR:

1. x³-x²-5x-1=0 (4U, 1985)
2. x³-6x²+24=0 (3U, 1977)
3. x³-x²-x-1=0 (3U, 1981)
4. x³+x-1=0 (3U, 2004)
5. x⁴-x-13=0 (3U, 1969)

See if you can do them.

<a href="http://www4.tpgi.com.au/nanahcub/answers.gif">Answers</a>

<a href="http://www4.tpgi.com.au/nanahcub/cubic.gif">Cubic formula</a>

<a href="http://mathworld.wolfram.com/CubicFormula.html">Proof of cubic formula</a>

<a href="http://www4.tpgi.com.au/nanahcub/quartic.gif">Quartic formula</a>

<a href="http://mathworld.wolfram.com/QuarticEquation.html">Proof of quartic formula</a>

Historical references:

Boyer, C. B. and Merzbach, U. C. A History of Mathematics, 2nd ed. New York: Wiley, pp. 282-287, 1991.

Deakin, M., History of Mathematics: The Invention of Complex Numbers, Parabola, 41 (2), pp. 13-20, 2005.

Deakin, M., History of Mathematics: Solving Cubic Equations, Parabola, 41 (3), pp. 39-45, 2005.