If (x^4+1/x^4)=34, then find value of (x-1/x). Please explain in detail.
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Answered by
89
This is required answer.
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TeachMeMath:
Hey can you explain how you got that -2? In similar questions I have seen people using +2.
Well I will explain from his side
There is an identity a² + b² = (a + b)² - 2ab
So he had just apply that
Thank you
Answered by
59
Given that,
x⁴ + 1/x⁴ = 34
Now we know that, if we will multiply x² with 1/x² we will get 1
x² × 1/x² = 1
Now add 2 × x² × 1/x² on both sides,
=> x⁴ + 1/x⁴ + 2x² × 1/x² = 34 + 2x² × 1/x²
Now if you see clearly you will find that,
x⁴ + 1/x4 + 2x² × 1/x² = (x² + 1/x²)²
as, a² + b² + 2ab = (a + b)²
=> (x² + 1/x²)² = 34 + 2x² × 1/x²
As we know, x² × 1/x² = 1
=> (x² + 1/x²)² = 34 + 2
=> (x² + 1/x²)² = 36
=> (x² + 1/x²) = √36
=> x² + 1/x² = 6
Again, subtract 2x × 1/x from both sides,
This time we get,
x² + 1/x² - 2x × 1/x = 6 - 2x × 1/x
=> (x - 1/x)² = 6 - 2
=> (x - 1/x)² = 4
=> x - 1/x = √4
=> x - 1/x = 2
Hope you understand
x⁴ + 1/x⁴ = 34
Now we know that, if we will multiply x² with 1/x² we will get 1
x² × 1/x² = 1
Now add 2 × x² × 1/x² on both sides,
=> x⁴ + 1/x⁴ + 2x² × 1/x² = 34 + 2x² × 1/x²
Now if you see clearly you will find that,
x⁴ + 1/x4 + 2x² × 1/x² = (x² + 1/x²)²
as, a² + b² + 2ab = (a + b)²
=> (x² + 1/x²)² = 34 + 2x² × 1/x²
As we know, x² × 1/x² = 1
=> (x² + 1/x²)² = 34 + 2
=> (x² + 1/x²)² = 36
=> (x² + 1/x²) = √36
=> x² + 1/x² = 6
Again, subtract 2x × 1/x from both sides,
This time we get,
x² + 1/x² - 2x × 1/x = 6 - 2x × 1/x
=> (x - 1/x)² = 6 - 2
=> (x - 1/x)² = 4
=> x - 1/x = √4
=> x - 1/x = 2
Hope you understand
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