# One number is 2 more than 2 times another. Their product is 2 more than 2 times their sum, how do you find the two integers?

Jul 5, 2015

Let's call the smaller number $x$.
Then the other number will be $2 x + 2$

#### Explanation:

Sum: $S = x + \left(2 x + 2\right) = 3 x + 2$
Product: $P = x \cdot \left(2 x + 2\right) = 2 {x}^{2} + 2 x$

$P = 2 \cdot S + 2$

Substituting:
$2 {x}^{2} + 2 x = 2 \cdot \left(3 x + 2\right) + 2 = 6 x + 4 + 2$

Everything to one side:
$2 {x}^{2} - 4 x - 6 = 0 \to$ divide everything by $2$
${x}^{2} - 2 x - 3 = 0 \to$ factorise:
$\left(x - 3\right) \left(x + 1\right) = 0 \to x = - 1 \mathmr{and} x = 3$

If we use the $2 x + 2$ for the other number, we get the pairs:
$\left(- 1 , 0\right) \mathmr{and} \left(3 , 8\right)$