Solve the equation by making an appropriate substitition. 
             x4 - 14x2 - 32 = 0 
Thanks for helping me.

The "literal part" of a term is the term without its numerical coefficient.

If the literal part of the first term happens to be the square of the literal
part of the second term, then we substitute w for the literal part of the
second term and w2 for the literal part of the first term.

The first term is 1x4, and so its literal part is just the x4 part without the
coefficient 1.

The second term is -14x2, and so its literal part is just the x2 part without
the coefficient -14.

Notice that x4 just happens to be the square of x2. so we substitute w2 for x4
and w for x2, so the original equation

        x4 - 14x2 - 32 = 0

becomes

         w2 - 14w - 32 = 0

To factor the left side we think of two numbers which have product -32 and
sum -14.  These are -16 and +2.  So the above factors as

       (w - 16)(w + 2) = 0

Then we use the zero factor property and set each factor = 0

Setting w - 16 = 0 gives w = 16

Setting w + 2 = 0 gives w = -2

However we don't want w, we want x.  So we now replace w by x2 in each


             w = 16                    w = -2                   
            x2 = 16                   x2 = -2__
             x = ±4                    x = ±Ö-2
                                       x = ±iÖ2
                                      _         _
So there are four solutions: 4, -4, iÖ2, and -iÖ2

Edwin
AnlytcPhil@aol.com