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Solutions to Problems; Chapter 1 259

Take u2 + v2 = 3x - 4. Then 2u2 = 2x - 3, 2v2 = 42: - 5. If 3: < 514, the

square roots are

If x > 312, the square roots are

Finally, it can be seen that, if 514 5 x 5 312, the square roots are

9.8. Let p(x) = ox2 + px + y and q(x) = Az2 + px + V. The condition

that the given quartic is of the form p(q(x)) leads to, on comparison of

coefficients,

u=crA2

b = 2(~Ap

c=‘LaXv+a~2+/3X

d= 2cY#Uv+pp

e = av2+/3v+-y.

The first two conditions lead to 2pu = Xb; the second and third lead to

cp - dX = op3. Eliminating ~1 from these two equations and noting that

a = oX2, we obtain the necessary condition

4abc - 8u2d = b3.

On the other hand, suppose that this condition is satisfied. If b = 0, then
d = 0, and we can takep(x) = ux2+cx+e, q(x) = z2. Suppose that b # 0.
Choose X = 1, (Y = a, p = b/2u. Any choice of p and v which satisfies
d = bv + (b/Su)/3 will give a correct expression for c. Finally, the equation
for e dictates the appropriate value of y. Thus, the coefficients of p and q
are found.
9.9. xu = u + v - uv = yv. Substituting v/u = x/y into the first pair of
equations yields the second pair.

9.10. The equation can be rewritten (x2 - 7x + 10)(x2 - 7x + 12) = 360,
which leads to (;~~-7x+ll)~ = 361. Hence, ~~-72-8 = 0 or z2-7x+30 =
0, leading to z = -1, 8, (l/2)(7 f i&f).

9.11. That x4y2+y4z2fz4x2 > 3x2y2z2 is a consequence of the arithmetic-
geometric mean inequality (Exercise 1.5.9). Suppose that the polynomial
is the sum of the squares of polynomials f(x, y, z). Each of these must