Unknown

46 1. Fundamentals

7.7. A complete collection of quadratic polynomials which can be ex-

pressed as the product of two linears can be obtained by multiplying

together all possible (not necessarily distinct) linear parts; the num-

ber of ways of doing this is easily determined.

8.1. Express tan(A + B) in terms of tan A and tan II, and thence in terms

of p and q.

8.2. Write the left side of the equation in the form ax2 + bz + c. Verify

that

3 k(i - 1)i = (n + l)n(n - 1).

i=l

8.3. Consider (r - s2)(s - r2), w h ere r and s are the roots. Whatever

method you use, be sure to show that the condition you obtain implies

and is implied by one root being the square of the other.

8.4. By considering q(t) = p(n + t), it suffices to prove the result is true

for n = 0 and any quadratic.

8.5. Express the discriminant of the second quadratic as the sum of a

square and a multiple of the discriminant of the first quadratic.

8.6. x2 + (b + 1)x + b has integer zeros for b E Z.

8.7. a and b have, respectively, the same sum and the same product as -c

and -d.

8.8. Substitute .z = r, the real root, into the equation and separate the

real and imaginary parts.

8.9. A common root of the two equations is a root of any equation of the

form f(x)(x” + px + 9) + g(x)(px2 + qx + 1) = 0.

8.11. tan(n cot x) = tan(z/2 - x tan x).

8.12. Form the discriminant and complete some obvious squares.

8.13. p(l), p(-1), p(i), p(-i) all belong to the unit disc; what does this
mean in terms of the coefficients a, b?

8.14. Solve the quadratic equation by completing the square. The line join-
ing the complex numbers r and s consists of the points (1 - t)r + ts
where t is real.

8.15. Let cos48 = (b2 - 8ac)/b2. Determine cos2 28 then take its square
root and find cos2 0.

8.16. Solve the equation for z by completing the square. Note that