ADVANCED EXERCISES

(5) Find a quadratic equation with integer coefficients having the following roots:

	9  5
        --------
	   16

Let p(x) be the required quadratic with integer coefficients.
Since each of the two roots gives a linear factor of p(x), we have

         p(x) = K[x - (9+5)/16][x - (9-5)/16]

              = (K/256)[16x - (9+5)][16x - (9-5)]

              = (K/256)[256x2 - 288x - 744]

So K can be any nonzero number such that K/256 is an integer and p(x) will
have integer coefficients. By construction, the original two numbers are
roots of p(x). If K = 256, we get the quadratic equation

                  256x2 - 288x - 744 = 0       (*)

We can make the coefficients even smaller integers by noting that each
coefficient in (*) is divisible by 8, which gives the equation:

		  32x2 - 36x - 93 = 0.