If x + y = 12 and xy = 32, find x² + y.

x + y = 12

⇒ y = 12-x …Equation (i)

xy = 32

Putting the value of y from above Equation (i) we get

x(12-x) = 32

⇒ 12x-x² = 32

⇒x²-12x + 32 = 0

Solving the above equation by the method of factorization we get

⇒ x²-8x-4x + 12 = 0

⇒x(x-8)-4(x-8) = 0

⇒(x-4)(x-8) = 0

x = 4,8

When x = 4, y = 8

⇒ x² + y = 24

When x = 8, y = 4

⇒ x² + y = 68