Question

if x+y = 5 , xy = 18, then value x3+y3​

Answer 1

Given:

x+y = 5 , xy = 18

To Find:

x³+y³​

Solution:

We will use the following algebric identity:-

(a +b)³ = a³ + b³ + 3ab(a + b)..........(i)

Rearranging the above equation, we get,

a³ + b³ = (a + b)³ - 3ab (a +b)

According to the question, x =a and y = b; subsituting these values in eq(i),

x³ +y³ = (x +y)³ - 3xy(x + y)..........(ii)

Since, x+y = 5 and xy = 18  (Given)

Putting these values in eq(ii), we get

    x³ +y³ = (5)³ - 3(18)(5)

⇒  x³ +y³ = (125 - 270) = - 145

Hence, the value of  x³ +y³ is -145.

Answer 2

SOLUTION

GIVEN

$\sf{x + y = 5 \: \: \: and \: \: xy = 18}$

TO DETERMINE

$\sf{ {x}^{3} + {y}^{ 3} }$

FORMULA TO BE IMPLEMENTED

We are aware of the identity that

$\sf{ {(x + y)}^{3} = {x}^{3} + 3 {x}^{2} y + 3x {y}^{2} + {y}^{3} }$

EVALUATION

Here it is given that

$\sf{x + y = 5}$

Cubing both sides we get

$\sf{ {(x + y)}^{3} = {(5)}^{3} }$

$\implies \: \sf{ {x}^{3} + 3 {x}^{2} y + 3x {y}^{2} + {y}^{3} = 125 }$

$\implies \: \sf{ {x}^{3} + {y}^{3} + 3 {x}^{2} y + 3x {y}^{2} = 125 }$

$\implies \: \sf{ {x}^{3} + {y}^{3} + 3xy(x + y) = 125 }$

$\implies \: \sf{ {x}^{3} + {y}^{3} +( 3 \times 18 \times 5) = 125 }$

$\implies \: \sf{ {x}^{3} + {y}^{3} +270 = 125 }$

$\implies \: \sf{ {x}^{3} + {y}^{3} = 125 - 270 }$

$\implies \: \sf{ {x}^{3} + {y}^{3} = - 145 }$

FINAL ANSWER

$\boxed{ \: \: \: \sf{ {x}^{3} + {y}^{3} = - 145 } \: \: }$

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