Prove that 3/2 root 5 is irrational

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step 1: 3 + 2 sqrt(5) is rational if and only if sqrt(5) is rational. This should be easy for you to show. 
step 2: we now show sqrt(5) is irrational. Suppose sqrt(5) = a/b. The 5b^2 = a^2. By unique factorization theorem, the both a and b have prime factorisations, say b = p_1^e_1*...*p_k^e_k and a = q_1^f_1 * .. * q_l^f_l , where the p_i are unique, same with q_j. The squaring results in doubling the exponents, so the right hand side has all even exponents. On the other hand, the left hand side has all even exponents except for the prime 5 which has odd. By unique factorization, this cannot happen: we cannot have an odd exponent on one side and all even exponents on the other. Therefore, it cannot be rational.

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12