Question

Find two positive numbers whose difference equals 8 and whose product equals 6.

Answer 1

The answer is x = 4 + √22, y = √22 - 4

x - y = 8
x * y = 6
______
y = x - 8
x * y = 6
______
x(x - 8) = 6
x² - 8x = 6
x² - 8x + 16 = 6 + 16
x² - 2 * x * 4 + 4² = 22

a² - 2 * a * b + b² = (a - b)²

(x - 4)² = 22
x - 4 = √22
x = 4 +- √22

x1 = 4 + √22 x2 = 4 - √22
y1 = x - 8 = 4 + √22 - 8 y2 = x - 8 = 4 - √22 - 8
y1 = - 4 + √22 y2 = - 4 - √22

Since y2 is negative number, and we need positive numbers, the answer is
x = 4 + √22
y = √22 - 4