The product of two numbers is 800. Their difference is 20. What are the two numbers?

The product of two numbers is 800.  The difference between the numbers is 20.  What are the two numbers?

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x = one number
y = the other number

xy = 800  {the product of two numbers is 800}
​x - y = 20  {their difference is 20}

Solve the second equation for x, by adding y to each side.
x - y = 20  {the second equation}
x = y + 20  {added y to each side}

Substitute (y + 20), in for x, into first equation.
xy = 800  {first equation}
(y + 20)y = 800  {substituted (y + 20) in for x}
y² + 20y = 800  {used distributive property}
y² + 20y - 800 = 0  {subtracted 800 from each side}
(y + 40)(y - 20) = 0  {factored into two binomials}
y + 40 = 0 or y - 20 = 0  {set each factor equal to zero}
y = -40 or y = 20  {solved each equation for y}

Substitute -40 and 20, back in for y, into first equation.
If y = -40
xy = 800  {first equation}
-40x = 800  {substituted -40 for y}
x = -20  {divided each side by -40}

The product of -20(-40) = 800
The difference of -20 - (-40) = -20 + 40 = 20  {this could be argued, based on the negatives}
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If y = 20
xy = 800 {first equation}
20x = 800  {substituted 20 in for y}
x = 40  {divided each side by 20}
The product of 20(40) = 800
the difference of 40 - 20 = 20

x = 40 and y = 20
or 
x = -20 and y = -40

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