Solve the following three problems. Write your solutions in English.
1. We will derive the quadratic equation by solving a system of linear equations. The equation ax2 + bx + c = 0 with a ≠ 0 has real solutions r, s if and only if ax2 + bx + c = a(x - r)(x - s). The calculation below shows that the factorization exists if and only if b2 - 4ac ≥ 0.
a) By equating coefficients of corresponding powers of x, obtain the equations r + s = -b/a and rs = c/a. Use these to prove that (r - s)2 = (b2 - 4ac)/a2.
b) From (a), obtain r + s = -b/a and r - s = √(b2 - 4ac)/a. Solve this system for r, s in terms of a, b, c.
c) What happens if the second equation in (b) is r - s = -√(b2 - 4ac)/a?
2. Given a real number x, let A be the statement "1/2 < x < 5/2", let B be the statement x ∈ Z, let C be the statement x2 = 1, and let D be the statement "x = 2". Which statements below are true for all x ∈ R?
a) A ⇒ C.
b) B ⇒ C.
c) (A ∧ B) ⇒ C.
d) (A ∧ B) ⇒ (C ∨ D).
e) C ⇒ (A ∧ B).
f) D ⇒ [A ∧ B ∧ (¬C)]. g) (A ∨ C) ⇒ B.
3. Let x, y be integers. Determine the truth value of each statement below.
a) xy is odd if and only if x and y are odd.
b) xy is even if and only if x and y are even.