Let a and b be real number that satisfy

a⁴ + a²b² + b⁴ = 900

a² + ab + b² = 45

Find the value of 2ab.

Answer.

Let a⁴ + a²b² + b⁴ = 900  ------- eq(1)

& a² + ab + b² = 45 ------- eq(2)

Simplify eq(1),

Add and Subtract a²b² for square completion,

(a²)² + 2(a²)(b²) + (b²)² - (ab)² = 900

(a² + b²)² - (ab)² = 900

(a² + b² + ab)(a² + b² - ab ) = 900 ------- eq(3)

We know that, a² + b² = 45 – ab ------- eq(4)

Put eq(4) in eq(3)

(45 - ab + ab)(45 - ab - ab ) = 900

(45 - 2ab)(45) = 900

45 - 2ab = 20

-2ab = -25

2ab = 25