What is the minimum value of the expression 2x^2 + y^2 + 8x - 24y + 62 for real x and y?
2x^2 + 8x + y^2 - 24y + 62 complete the square on x, y
2(x^2 + 4x + 4) + (y^2 - 24y + 144) + 62 - 8 - 144
2 (x + 2)^2 + (y - 12)^2 - 90
The min of 90 occurs when x = -2 and y = 12
