Question

Solve this equation over the reals:

Answer 1

• x = -3, y = - 2

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Given equation:

• (x² + 6x + 17)(y² + 4y + 7) = 24

Complete the square in both parenthesis:

• [(x² + 6x + 9) + 8][(y²+ 4y + 4) + 3] = 24
• [(x + 3)² + 8][(y + 2)² + 3] = 24

We know that the square of a rational number is a positive number or zero but never negative. Then:

• (x + 3)² + 8 ≥ 8 and
• (y + 2)² + 3 ≥ 3

The minimum value of the product is 8*3 = 24 if both squares are equal to zero:

• (x + 3)² = 0 ⇒ x + 3 = 0 ⇒ x = - 3
• (y + 2)² = 0 ⇒ y + 2 = 0 ⇒ y = - 2