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
Answers
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.)
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