# How do you find the two consecutive even integers whose product is 840?

Jul 5, 2016

Translate the problem to an algebraic statement and solve a quadratic equation to find that there are two pairs of numbers that satisfy the problem.

#### Explanation:

When we are solving algebraic problems, the first thing we must do is define a variable for our unknowns. Our unknowns in this problem are two consecutive even numbers whose product is $840$. We'll call the first number $n$, and if they're consecutive even numbers, the next one will be $n + 2$. (For example, $4$ and $6$ are consecutive even numbers and $6$ is two more than $4$).

We are told that the product of these numbers is $840$. That means these numbers, when multiplied together, produce $840$. In algebraic terms:
$n \cdot \left(n + 2\right) = 840$

Distributing the $n$, we have:
${n}^{2} + 2 n = 840$

Subtracting $840$ from both sides gives us:
${n}^{2} + 2 n - 840 = 0$

Now we have a quadratic equation. We can try to factor it, by finding two numbers that multiply to $- 840$ and add to $2$. It might take a while, but eventually you'll find these numbers are $- 28$ and $30$. Our equation factors into:
$\left(n - 28\right) \left(n + 30\right) = 0$

Our solutions are:
$n - 28 = 0 \to n = 28$
$n + 30 = 0 \to n = - 30$

Thus, we have two combinations:

• $28$ and $28 + 2$, or $30$. You can see that $28 \cdot 30 = 840$.
• $- 30$ and $- 30 + 2$, or $- 28$. Again, $- 30 \cdot - 28 = 840$.
Jul 5, 2016

The reqd. nos. are $- 30 , - 28$ or, $28 , 30.$

#### Explanation:

Suppose that the reqd. integers are $2 x$ and $2 x + 2$

By given, then, we have $2 x \cdot \left(2 x + 2\right) = 840 \Rightarrow 4 x \left(x + 1\right) = 840$.

$\therefore {x}^{2} + x = \frac{840}{4} = 210 ,$ or, ${x}^{2} + x - 210 = 0$

$\therefore {x}^{2} + 15 x - 14 x - 210 = 0$

$\therefore x \left(x + 15\right) - 14 \left(x + 15\right) = 0$

$\therefore \left(x + 15\right) \left(x - 14\right) = 0$
$\therefore x = - 15 , \mathmr{and} , x = 14$

CASE I

$x = - 15$, the reqd. nos. are $2 x = - 30 , 2 x + 2 = - 28.$

Case II

$x = 14$, the. nos. are $2 x = 28 , 2 x + 2 = 30$