Question

if x² + y² = 29 and xy = 2 , Find the value of , (1) x+y , (2) x-y , (3) x⁴ + y⁴​

Answer 1

Given that, x² + y² = 29 and xy = 2.

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We'll have to find the value of :

⑴ x + y

⑵ x – y

⑶ x⁴ + y⁴

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Finding x + y using (x + y)² formula :

➯ (x + y)² – 2xy = x² + y²

⇒ (x + y)² – 2 (2) = 29

⇒ (x + y)² – 4 = 29

⇒ (x + y)² = 29 + 4

⇒ (x + y)² = 33

⇒ x + y = ±√33

Finding x – y using (x – y)² formula :

➯ (x – y)² = x² + y² – 2xy

⇒ (x – y)² = 29 – 2 (2)

⇒ (x – y)² = 29 – 4

⇒ (x – y)² = 25

⇒ x – y = √25

⇒ x – y = ±5

Finding x⁴ + y⁴ by squaring on both sides :

➯ (x² + y²)² = (29)²

⇒ x⁴ + y⁴ + 2x²y² = 841

⇒ x⁴ + y⁴ + 2 (xy)² = 841

⇒ x⁴ + y⁴ + 2 (2)² = 841

⇒ x⁴ + y⁴ + 2 (4) = 841

⇒ x⁴ + y⁴ + 8 = 841

⇒ x⁴ + y⁴ = 841 – 8

⇒ x⁴ + y⁴ = 833

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Therefore, we get :

⑴ x + y = ±√33

⑵ x – y = ±5

⑶ x⁴ + y⁴ = 833