lve the following questions. (Any FOUR)
If a and B are the roots of equation. x* + 5x - 1 = 0 the find the value of a3+b3
Answers
Answer:
- 140
Step-by-step explanation:
Given-----> If α and β are the roots of the equation x² + 5x - 1 = 0
To find------> Value of α³ + β³
Solution------> ATQ ,
x² + 5x - 1 = 0
Comparing it with ax² + bx + c = 0 , we get,
a = 1 , b = 5 , c = -1
ATQ, roots of given equation is , α and β
We know that ,
Sum of roots = - Coefficient of x / Coefficient of x²
=> α + β = - b / a
=> α + β = - 5 / 1
=> α + β = - 5
Product of roots
= Constant term / Coefficient of x²
=> α β = - 1 / 1
=> α β = - 1
We know that,
( a + b )³ = a³ + b³ + 3ab ( a + b )
=> a³ + b³ = ( a + b )³ - 3ab ( a + b )
Now putting a = α and b = β , we get,
=> α³ + β³ = ( α + β )³ - 3αβ ( α + β )
= (- 5 )³ - 3 (- 1 ) (- 5 )
= ( - 125 ) - 15
=> α³ + β³ = - 140
ATQ ,
x² + 5x - 1 = 0
Comparing it with ax² + bx + c = 0 , we get,
a = 1 , b = 5 , c = -1
ATQ, roots of given equation is , α and β
We know that ,
Sum of roots = - Coefficient of x / Coefficient of x²
=> α + β = - b / a
=> α + β = - 5 / 1
=> α + β = - 5
Product of roots
= Constant term / Coefficient of x²
=> α β = - 1 / 1
=> α β = - 1
We know that,
( a + b )³ = a³ + b³ + 3ab ( a + b )
=> a³ + b³ = ( a + b )³ - 3ab ( a + b )
Now putting a = α and b = β , we get,
=> α³ + β³ = ( α + β )³ - 3αβ ( α + β )
= (- 5 )³ - 3 (- 1 ) (- 5 )
= ( - 125 ) - 15
=> α³ + β³ = - 140
hope it helps you