Question

If x + y + z= 10 and x^2+y^2+z^2=40 , then find the value of (xy+yz+zx) ​

Answer 1

Given:

• x + y + z = 10

• x² + y² + z² = 40

To find out:

Find the value of ( xy + yz + zx )

Solution :

We have,

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

➠ ( 10 )² = x² + y² + z² + 2 ( xy + yz + zx )

➠ 100 = 40 + 2 ( xy + yz + zx )

➠ 100 - 40 = 2 ( xy + yz + zx )

➠ 60 = 2 ( xy + yz + zx )

➠ xy + yz + zx = 60/2

➠ xy + yz + zx = 30

Hence, the value of ( xy + yz + zx ) is 30.

Additional information:

• ( a + b )² = a² + b² + 2ab

• ( a - b )² = a² + b² - 2ab

• ( a + b )³ = a³ + b³ + 3ab ( a + b )

• ( a - b )³ = a³ - b³ - 3ab ( a - b )

• a³ + b³ = ( a + b ) ( a² - ab + b² )

Answer 2

Given

x + y + z = 10

x² + y² + z² = 40

To find

Value of (xy + yz + zx)

Solution

Here,given the value of (x + y + z) = 10 & (x² + y² + z²) = 40

We know identity :

➥ (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Putting values we get :

➝ (10)² = 40 + 2(xy + yz + zx)

➝ 100 = 40 + 2(xy + yz + zx)

➝ 100 - 40 = 2(xy + yz + zx)

➝ 60 = 2(xy + yz + zx)

➝ xy + yz + zx = 60/2

➝ (xy + yz + zx) = 30

Therefore,

Required value of (xy + yz + zx) = 30

_________________________________

Some more identities :

⟼ (a + b + c)³ = a³ + b³ + c³ + 3(a + b)(b + c)(c + a)

⟼ (x + a)(x + b) = x² + (a + b)x + ab

⟼ a² - b² = (a + b)(a - b)

⟼ 4ab = (a + b)² - (a - b)²

⟼ 2(a² + b²) = (a + b)² + (a - b)²