Answer:
Given, x + 1/x = 2
Squaring both sides, we get:
(x + 1/x)² = 2²
x² + 1/x² + 2 = 4
x² + 1/x² = 2
Now, multiplying both sides of x + 1/x = 2 by x, we get:
x² + 1 = 2x
x³ + x = 2x²
Dividing both sides by x, we get:
x² + 1/x = 2x
(x² + 1/x)² = (2x)²
x⁴ + 1/x² + 2 = 4x²
x⁴ + 1/x² = 4x² - 2
Substituting the value of x² + 1/x² from the first equation, we get:
x³ + 1/x³ + x + 1/x = 4x² - 2
x³ + 1/x³ = 4x² - 2 - 2 = 4x² - 4
Multiplying both sides of x³ + 1/x³ = 4x² - 4 by x, we get:
x⁴ + 1/x² = 4x³ - 4x
Substituting the value of x⁴ + 1/x² from the second equation, we get:
x³ + 1/x³ = 4x² - 4 - 4x³ + 4x
x³ + 1/x³ = -4x³ + 4x² + 4x - 4
Adding 4 to both sides, we get:
x³ + 1/x³ + 4 = -4x³ + 4x² + 4x
x³ + 1/x³ + 4 = 4(-x³ + x² + x)
Dividing both sides by 4, we get:
x³/4 + 1/(4x³) + 1 = -x³ + x² + x
Multiplying both sides by x, we get:
x⁴/4 + 1/(4x²) + x = -x⁴ + x³ + x²
x³ + 1/x³ = x² - 2
x³ + 1/x³ = (x + 1/x)² - 2
x³ + 1/x³ = 2² - 2
x³ + 1/x³ = 2
Therefore, x⁴ + 1/x⁴ = (x² + 1/x²)² - 2 = 2² - 2 = 2
Hence, proved.
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Verified answer
Answer:
Given, x + 1/x = 2
Squaring both sides, we get:
(x + 1/x)² = 2²
x² + 1/x² + 2 = 4
x² + 1/x² = 2
Now, multiplying both sides of x + 1/x = 2 by x, we get:
x² + 1 = 2x
x³ + x = 2x²
Dividing both sides by x, we get:
x² + 1/x = 2x
Squaring both sides, we get:
(x² + 1/x)² = (2x)²
x⁴ + 1/x² + 2 = 4x²
x⁴ + 1/x² = 4x² - 2
Substituting the value of x² + 1/x² from the first equation, we get:
x³ + 1/x³ + x + 1/x = 4x² - 2
x³ + 1/x³ = 4x² - 2 - 2 = 4x² - 4
Multiplying both sides of x³ + 1/x³ = 4x² - 4 by x, we get:
x⁴ + 1/x² = 4x³ - 4x
Substituting the value of x⁴ + 1/x² from the second equation, we get:
x³ + 1/x³ = 4x² - 4 - 4x³ + 4x
x³ + 1/x³ = -4x³ + 4x² + 4x - 4
Adding 4 to both sides, we get:
x³ + 1/x³ + 4 = -4x³ + 4x² + 4x
x³ + 1/x³ + 4 = 4(-x³ + x² + x)
Dividing both sides by 4, we get:
x³/4 + 1/(4x³) + 1 = -x³ + x² + x
Multiplying both sides by x, we get:
x⁴/4 + 1/(4x²) + x = -x⁴ + x³ + x²
Substituting the value of x⁴ + 1/x² from the second equation, we get:
x³ + 1/x³ = x² - 2
Substituting the value of x² + 1/x² from the first equation, we get:
x³ + 1/x³ = (x + 1/x)² - 2
x³ + 1/x³ = 2² - 2
x³ + 1/x³ = 2
Therefore, x⁴ + 1/x⁴ = (x² + 1/x²)² - 2 = 2² - 2 = 2
Hence, proved.