The difference of the squares of two distinct positive numbers is equal to twice the square of their difference. What is the ratio of the smaller number to the larger?

1:3

Step-by-step explanation:

The difference of the squares of two distinct positive numbers is equal to twice the square of their difference. What is the ratio of the smaller number to the larger? I'll call the numbers a and b.

a² - b² = 2 (a-b) ²

a² - b² = 2 (a-b) (a-b) First Outside Inside Last (a-b) (a-b)

a² - b² = 2 (a²-ab-ab+b²) = 2 (a²-2ab+b²) = 2a² - 4ab + 2b²

a² - b² = 2a² - 4ab + 2b² subtract a² - b² from both sides

-a² + b² - a² + b²

0 = a² - 4ab + 3b² factor into multiplication of two binomials. What factors (two

numbers multiply) to 3 add up to - 4? The only two numbers

are - 3 and - 1.

0 = (a-3b) (a-1b) set each binomial equal to zero and solve

0 = a-3b or 0 = a-1b

+3b + 3b or + 1b + 1b

3b = a or 1b = a

1:3 or 1:1 (but 1:1 says they are the same number, so if there is a smaller number, only 1:3 is correct)