Question

●If x^3+1/x^3=110,then x+1/x=????

●If x^3-1/x^3=14,then x-1/x=????

Plz help (×_×)
I forgot how to do this.
#ExamFever

Answer 1

if x³ + 1/x³ = 110, then x + 1/x = ?

use formula, a³ + b³ = (a + b)³ - 3ab(a + b)

so, x³ + 1/x³ = (x + 1/x)³ - 3x × 1/x(x + 1/x)

110 = (x + 1/x)³ - 3(x + 1/x)

Let (x + 1/x) = P

so, 110 = P³ - 3P

or, P³ - 3P - 110 = 0

or, P³ - 5P² + 5P² - 25P + 22P - 110 = 0

or, P²(P - 5) + 5P(P - 5) + 22(P - 5) = 0

or, (P - 5)(P² + 5P + 22) = 0

hence, P = 5 , P² + 5P + 22 ≠ 0

hence, P = x + 1/x = 5

again, x³ - 1/x³ = 14 , then x - 1/x = ?

x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

14 = (x - 1/x)³ + 3(x - 1/x)

Let (x - 1/x) = Q

so, 14 = Q³ + 3Q

or, Q³ + 3Q - 14 = 0

or, Q³ - 2Q² + 2Q² - 4Q + 7Q - 14 = 0

or, Q²(Q - 2) + 2Q(Q - 2) + 7(Q - 2) = 0

or, (Q - 2)(Q² + 2Q + 7) = 0

or, Q = 2 , Q² + 2Q + 7 ≠ 0

hence, Q = x - 1/x = 2