Question

if x+y+z=5 and xy+yz+xz=10, then find the value of x^3+y^3+z^3-3xyz.​

Answer 1

${ \huge{ \underline{ \underline{ \sf{ \green{GivEn : }}}}}}$

• x + y + z = 5

• xy + yz + zx = 10

${ \huge{ \underline{ \underline{ \sf{ \green{To \: find :}}}}}}$

• The value of x³ + y³ + z³ - 3xyz

Formula to be used :-

• x³ + y³ + z³ - 3xyz

= ( x + y + z) ( x² + y² + z² - xy - yz - zx)

• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz

${ \huge{ \underline{ \underline{ \sf{ \green{SoluTion : }}}}}}$

At first, we need to find the value of x² + y² + z²

We know,

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz

⟶ x² + y² + z² = (x + y + z)² -( 2xyb + 2yz + 2xz)

⟶ x² + y² + z² = 5² -2(10)

⟶ x² + y² + z² = 25 - 20

⟶ x² + y² + z² = 5

Hence, value of x² + y² + z² is = 5.

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Now,

x³ + y³ + z³ - 3xyz

Put the given values in the formula

= ( x + y + z) ( x² + y² + z² - xy - yz - zx)

= 5 {5 - ( 10) }

= -25

Hence, value of x³ + y³ + z³ - 3xyz is = -25