If x⁴ – 83x² +1 = 0, then a value
of x³ -x-³ is :
Answers
x³ - x⁻³ = ± 756 if x⁴ – 83x² +1 = 0
Step-by-step explanation:
x⁴ – 83x² +1 = 0
Dividing both sides by x²
=> x² - 83 + 1/x² = 0
=> x² + 1/x² = 83
=> (x - 1/x)² + 2x(1/x) = 83
=> (x - 1/x)² + 2 = 83
=> (x - 1/x)² = 81
=> x - 1/x = ±9
Cubing both sides
=> (x - 1/x)³ = (±9)³
=> x³ - 1/x³ - 3x(1/x)(x - 1/x) = ±729
=> x³ - 1/x³ - 3(±9) = ±729
=> x³ - 1/x³ = ±729 ± 27
=> x³ - 1/x³ = ± 756
=> x³ - x⁻³ = ± 756
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value of x³ - x-³ = 756 or, -756
it is given that x⁴ - 83x² + 1 = 0, then we have to find value of x³ - x-³ or, x³ - 1/x³
let's resolve the x⁴ - 83x² + 1 = 0
dividing by x² from both sides,
⇒x⁴/x² - 83x²/x² + 1/x² = 0
⇒x² - 83 + 1/x² = 0
⇒(x² + 1/x²) = 83
using formula, a² + b² = (a - b)² + 2ab
⇒ (x + 1/x)² +2(x)(1/x) = 83
⇒(x - 1/x)² + 2 = 83
⇒(x - 1/x)² = 81 = 9²
⇒(x - 1/x) = ±9 ........(1)
now using formula, a³ - b³ = (a - b)³ + 3ab(a - b)
so, x³ - 1/x³ = (x - 1/x)³ + 3(x)(1/x)(x - 1/x)
= (x - 1/x)³ + 3(x - 1/x)
from equations (1),
= (±9)³ + 3(±9)
= ±729 ± 27
= ±756
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