Question 30560
<pre><b><font size = 3>Solve the equation by making an appropriate substitition. 
             x<sup>4</sup> - 14x<sup>2</sup> - 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 w<sup>2</sup> for the literal part of the first term.

The first term is 1x<sup>4</sup>, and so its literal part is just the x<sup>4</sup> part without the
coefficient 1.

The second term is -14x<sup>2</sup>, and so its literal part is just the x<sup>2</sup> part without
the coefficient -14.

Notice that x<sup>4</sup> just happens to be the square of x<sup>2</sup>. so we substitute w<sup>2</sup> for x<sup>4</sup>
and w for x<sup>2</sup>, so the original equation

        x<sup>4</sup> - 14x<sup>2</sup> - 32 = 0

becomes

         w<sup>2</sup> - 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 x<sup>2</sup> in each


             w = 16                    w = -2                   
            x<sup>2</sup> = 16                   x<sup>2</sup> = -2__
             x = ±4                    x = ±<font face = "symbol">Ö</font>-<u>2</u>
                                       x = ±i<font face = "symbol">Ö</font>2
                                      _         _
So there are four solutions: 4, -4, i<font face = "symbol">Ö</font>2, and -i<font face = "symbol">Ö</font>2

Edwin
AnlytcPhil@aol.com</pre>