if x + 1/x = 6 find x^3 - 1/x^3
Answers
Answer:
It is given that x−
x
1
=6
(i)
Cube both sides of the given equation,
(x−
x
1
)
3
=6
3
x
3
−
x
3
1
−3×x×
x
1
(x−
x
1
)=216
x
3
−
x
3
1
−3(6)=216
x
3
−
x
3
1
−18=216
x
3
−
x
3
1
=234
(ii)
Square both sides of the given equation,
(x−
x
1
)
2
=6
2
x
2
+
x
2
1
−2×x×
x
1
=36
x
2
+
x
2
1
−2=36
x
2
+
x
2
1
=38
EXPLANATION.
⇒ (x + 1/x) = 6.
As we know that.
Squaring on both sides of the equation, we get.
⇒ (x + 1/x)² = (6)².
⇒ x² + 1/x² + 2(x)(1/x) = 36.
⇒ x² + 1/x² + 2 = 36.
⇒ x² + 1/x² = 36 - 2.
⇒ x² + 1/x² = 34.
As we know that,
Formula of :
⇒ (x - 1/x)² = x² + 1/x² - 2(x)(1/x).
⇒ (x - 1/x)² = x² + 1/x² - 2.
Put the value of x² + 1/x² = 34 in the equation, we get.
⇒ (x - 1/x)² = 34 - 2.
⇒ (x - 1/x)² = 32.
⇒ (x - 1/x) = √32.
⇒ (x - 1/x) = 4√2.
As we know that,
⇒ (x - 1/x) = 6.
Cubing on both sides of the equation, we get.
⇒ (x - 1/x)³ = (6)³.
⇒ x³ - 3(x)²(1/x) + 3(x)(1/x)² - 1/x³ = 216.
⇒ x³ - 1/x³ - 3x + 3/x = 216.
⇒ x³ - 1/x³ - 3(x - 1/x) = 216.
Put the value of x - 1/x = 4√2 in the equation, we get.
⇒ x³ - 1/x³ - 3(4√2) = 216.
⇒ x³ - 1/x³ - 12√2 = 216.
⇒ x³ - 1/x³ = 216 + 12√2.