Question

If x^3 + 1/x^3 = 2, find x + 1/x.

Answer 1

Step-by-step explanation:

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Answer 2

EXPLANATION.

⇒ (x³ + 1/x³) = 2.

As we know that,

We can write equation as,

⇒ (x + 1/x)³ = x³ + 3(x)²(1/x) + 3(x)(1/x)² + (1/x)³.

⇒ (x + 1/x)³ = x³ + 1/x³ + 3x + 3/x.

Put the value of (x³ + 1/x³ = 2) in the equation, we get.

⇒ (x + 1/x)³ = 2 + 3(x + 1/x).

⇒ (x + 1/x)³ - 3(x + 1/x) - 2 = 0.

Let, we assume that,

⇒ (x + 1/x) = a.

⇒ a³ - 3a - 2 = 0.

Put the value of a = - 1 in the equation, we get.

⇒ (-1)³ - 3(-1) - 2 = 0.

⇒ - 1 + 3 - 2 = 0.

⇒ 3 - 3 = 0.

As, we can see that,

(a + 1) is a factor of equation : a³ - 3a - 2 = 0.

Now, divide the equation by : a + 1, we get.

⇒ (a + 1)[a² - a - 2] = 0.

⇒ (a + 1)[a² - 2a + a - 2] = 0.

⇒ (a + 1)[a(a - 2) + 1(a - 2)] = 0.

⇒ (a + 1)(a + 1)(a - 2) = 0.

⇒ (a + 1)²(a - 2) = 0.

⇒ a = - 1  and  a = 2.

For all real values of x,

⇒ (x + 1/x) = a.

⇒ (x + 1/x) = 2.