Question

if x^4+2x^2+1=5x^2....then what will bs the value of x+1/x

plz give the ans. with step by step explanation ​

Answer 1

Sᴏʟᴜᴛɪᴏɴ :-

→ x⁴ + 2x² + 1 = 5x²

→ x⁴ + 2x² - 5x² + 1 = 0

→ x⁴ - 3x² + 1 = 0

Taking x² common From LHS now,

→ x²(x² - 3 + 1/x²) = 0

→ x² + 1/x² - 3 = 0

→ x² + 1/x² = 3

Adding 2 Both sides Now, we get,

→ x² + 1/x² + 2 = 3 + 2

→ x² + 1/x² + 2 * x * 1/x = 5

comparing the LHS part with a² + b² + 2ab = (a + b)² Now, we get,

→ (x + 1/x)² = 5

Square - Root Both sides now, we get,

→ (x + 1/x) = ±√5 (Ans.)

Answer 2

Qᴜᴇsᴛɪᴏɴ:-

x⁴ + 2x² + 1 = 5x², find x + 1/x

Fᴏʀᴍᴜʟᴀᴇ Usᴇᴅ:-

(a + b)² = a² + b² + 2ab

Aɴsᴡᴇʀ:-

x⁴ + 2x² + 1 = 5x²

Take x² common on both sides

➮x²(x² + 2 + 1/x²) = x²(5)

Cancel x² on both sides

➮x² + 2 + 1/x² = 5

➮x² + 1/x² = 5 - 2

➮x² + 1/x² = 3-----------------Equation❶

(x + 1/x)² = x² + (1/x)² + (2 * x * 1/x) [By (a + b)² formula]

➮(x + 1/x)² = x² + 1/x² + 2

➮(x + 1/x)² = 3 + 2 [From ❶]

➮(x + 1/x)² = 5

➮(x + 1/x) = ±√5

Ans. = ±√5