Exercise 3

Let x be a real number. Solve the equation 3^2x-2(3^x)+1=0.

Exercise 4

Prove the following identities, where p, q, x, y, m, and n are real numbers:

8(p-q)+4(p+q)=2(p+3q)+10(p-q)

x(m-n)+y(n+m)=m(x+y)+n(y-x)

(x+3)(x+8)-(x-6)(x-4)=21x

m⁸-1=(m²+1)(m²-1)(m⁴+1)

Extra credit:

Let us consider a floor covered with square tiles; all the tiles have the same size, with Width=Height=A. We drop a coin of radius R on the floor. What is the probability that the coin falls on top of (at least) one line that delimits a tile? We assume such a line to be infinitesimally thin.

Let x be a real number. Solve the equation 32x-2(3")+1 =0. Exercise 4 Prove the following identities, where p, q, x, y, m, and n are real numbers: a) 8(p-q)+4(p+q)=2(p+3q)+10(pq) 5) x(m-n)+y(n+m)=m(x+y)+n(y-x) c) (x+3)(x+8)(x6)(x4)=21x d) m8-1=(m2+1)(m2-1)(m4+1) Extra credit: Let us consider a oor covered with square tiles; all the tiles have the same size, with Width=Height=A. We drop a coin of radius R on the oor. What is the probability that the coin falls on top of (at least) one line that deliInits a tile? We assume such a line to be innitesimally thin