Question

plz some one help me in this question
I will mark u as brainliest.... ​

Answer 1

    x² + 4x - 1 = 0

Subtract 4x from both sides.

⇒  x² - 1 = - 4x

Divide both sides by x.

⇒  (x² - 1) / x = - 4x / x

⇒  x - 1/x = - 4

Now take the square of both sides.

⇒  (x - 1/x)² = (- 4)²

⇒  x² + 1/x² - 2 = 16     →     (1)

Add 4 to both sides.

⇒  x² + 1/x² - 2 + 4 = 16 + 4

⇒  x² + 1/x² + 2 = 20

Factorise LHS.

⇒  (x + 1/x)² = 20

From this, we get,

⇒  x + 1/x = √20 = 2√5

And from (1),

    x² + 1/x² - 2 = 16

⇒  x² + 1/x² = 16 + 2 = 18

Answer 2

Question :-

If x² + 4x - 1 = 0 then find the value of

(i) x + 1/x

(ii) x² + 1/x²

Answer :-

(i) x + 1/x = 2√5

(ii) x² + 1/x² = 18

Solution :-

We can find value of x + 1/x from x² + 4x - 1 = 0.

x² + 4x - 1 = 0

⇒ x² + 4x = 1

⇒ x(x + 4) = 1

⇒ x + 4 = 1/x

⇒ x + 4 - 1/x = 0

⇒ x - 1/x = - 4

We know that

(a + b)² = (a - b)² + 4ab

Here a = x, b = 1/x

By substituting the values

⇒ (x + 1/x)² = (x - 1/x)² + 4(x)(1/x)

⇒ (x + 1/x)² = (x - 1/x)² + 4

⇒ (x + 1/x)² = (-4)² + 4

⇒ (x + 1/x)² = 16 + 4

⇒ (x + 1/x)² = 20

⇒ x + 1/x = √20

⇒ x + 1/x = √4 * √5

⇒ x + 1/x = 2(√5)

⇒ x + 1/x = 2√5

Squaring on both sides

(x + 1/x)² = (2√5)²

We know that

⇒ (x)² + (1/x)² + 2(x)(1/x) = 4(5)

[Since (a + b)² = a² + b² + 2ab]

⇒ x² + 1²/x² + 2 = 20

⇒ x² + 1/x² + 2 = 20

⇒ x² + 1/x² = 20 - 2

⇒ x² + 1/x² = 18