Unknown

150 4. Equations

5. For which values of a, b, c does the equation

x+a&+b+&=c

have infinitely many real zeros.

6. Let k be a positive real number. Solve for real x.

,/x+Jm=c.

7. Let a, b be positive. Solve the equation

d2ab + 2ax + 26x - $-b2-$= &=&d=-.

8. Solve (x2 + 3x + 2)(x2 + 7x + 12) + (x2 + 5x - 6) = 0.


9. (a) Let p(t) = at2 + bt + c. Suppose that u # v and that p(u) = u,
   p(v) = v. Let q(t) = p(p(t))- t. Show that u and v are two zeros
   of the quartic polynomial q(t) and determine a quadratic whose
   zeros are the other two zeros of q(t).
   (b) Apply (a) to solve

(t2 - 3t + 2)2 - 3(P - 3t + 2) + 2 -t = 0.

10. Solve for real 2:


11. Show that 4cos2(7r/14) is the greatest root of the equation

x3 - 7x2 + 14x - 7 = 0.

12. Consider the equation x4 = (1 - x)(1 - x2)2. Show that if either
    1 - x2 = -x3 or 1 - x2 = x holds, then the equation is satisfied.
    Deduce that

x4 - (1 - x)(1 - x2)2 = (x” - x2 + 1)(x2 + x - 1).

13. Find a real solution to the equation

(X2 - 9x - l)lO + 99x1° = loxs(x2 - 1).