Number Problem

The difference between two numbers is 3.  If the square of the smaller number is equal to 4 times the larger number, find the numbers.

x = larger number
y = smaller number 

x - y = 3   {difference between two numbers is 3}
y² = 4x   {square of smaller is equal to 4 times the larger} 

y = x - 3  {added y and subtracted 3 from each side in top original equation} 

(x - 3)² = 4x   {substituted (x - 3), in for y, into bottom original equation}
x² - 6x + 9 = 4x   {squared the binomial by multiplying by itself}
x² - 10x + 9 = 0   {subtracted 4x from each side}
(x - 9)(x - 1) = 0   {factored into two binomials}
x - 9 = 0 or x - 1 = 0   {set each factor equal to 0}
x = 9 or x = 1   {solved each equation for x} 

y = x - 3
y = 9 - 3 or y = 1 - 3 {substituted 9 and 1 for x into y = x - 3}
y = 6 or y = -2 

9 and 6
or 
1 and -2 are the sets of numbers

- Algebra House