Question

Two positive numbers have a difference of 8 and a product of 33. What are these numbers?

Answer 1

Well lets set up the equations

\[\large x - y = 8\]

\[\large x \times y = 33\]

Lets solve that top equation for 'x' by adding y to both sides

\[\large x = y + 8\]

Substitute that in for 'x' in the second equation
\[\large (y + 8)y = 33\]
Distribute
\[\large y^2 + 8y = 33\]
Lets complete the square there
\[\large (y + 4)^2 = 49\]
Take the square root of both sides
\[\large y + 4 = \sqrt{49}\]
Simplify
\[\large y + 4 = 7\]
Subtract 4 from both sides
\[\large y = 3\]

Now if we know y = 3 we can substitute that into the original equations to solve for 'x'

\[\large x - y = 8\]
\[\large x \times y = 33\]
Lets put that y = 3 in
\[\large x - 3 = 8\]
Add 3 to both sides
\[\large x = 11\]

Now lets check with that second equation

\[\large 3 \times 11 = 33 \checkmark \]