Question

If the product of two consecutive even integers is 288, how do you find the integers?

Answer 1

Hint: In this question we have to find the two consecutive integers whose product is 288, let the two numbers be $x$ and $x + 2$, and make a mathematical expression according to the question given and then we will get a quadratic equation and solve the equation using quadratic formula to get the required numbers.

Complete step by step solution:
Given that the product of two consecutive even integers is 288,
Let the required numbers be .. and $x + 2$,
Now the required mathematical expression will be ,
$ \Rightarrow x\left( {x + 2} \right) = 288$,
Now multiplying to simplify we get,
$ \Rightarrow {x^2} + 2x = 288$,
Now subtracting 288 on both sides of the equation we get,
$ \Rightarrow {x^2} + 2x - 288 = 288 - 288$,
Now simplifying we get,
$ \Rightarrow {x^2} + 2x - 288 = 0$,
Now this is a quadratic equation and this can be solved by quadratic formula which is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$,
Now here $a = 1$, $b = 2$ and $c = - 288$, by substituting the values in the formula we get,
$ \Rightarrow $$x = \dfrac{{ - 2 \pm \sqrt {{2^2} - 4\left( 1 \right)\left( { - 288} \right)} }}{{2\left( 1 \right)}}$,
Now simplifying we get,
$ \Rightarrow $$x = \dfrac{{ - 2 \pm \sqrt {4 - \left( { - 1152} \right)} }}{2}$,
Now simplifying we get,
$ \Rightarrow $$x = \dfrac{{ - 2 \pm \sqrt {4 + 1152} }}{2}$,
Now further simplifying we get,
$ \Rightarrow $$x = \dfrac{{ - 2 \pm \sqrt {1156} }}{2}$,
Now taking out the square root we get,
$ \Rightarrow $$x = \dfrac{{ - 2 \pm 34}}{2}$,
Now we get the two values for $x$ they are,
$ \Rightarrow x = \dfrac{{ - 2 + 34}}{2} = \dfrac{{32}}{2} = 16$ and,
$ \Rightarrow x = \dfrac{{ - 2 - 34}}{2} = \dfrac{{ - 36}}{2} = - 18$,
Let’s take $x = 16$, so one number is 16 and then the second number is $16 + 2 = 18$, and the product of the numbers is $16 \times 18 = 288$, which is what is given in the question.
Now let’s take $x = - 18$, so the number in -18 and then the second number is $ - 18 + 2 = - 16$, and the product of the numbers is $ - 18 \times - 16 = 288$, which is what is given in the question.
So, the numbers can either be 16 and 18 or the numbers can be -18 and -16.

$\therefore $ The consecutive even integers can either be 16 and 18 or -18 and -16 whose product is 288.

Note:
Consecutive integers are integers that follow in a fixed sequence, each number being 1 more than the previous numbers, and they are represented as n, n+1, n+2….., where n is any integer.
There are three types of consecutive integers, they are,
1. Normal consecutive integers.
2. Even consecutive integers.
3. Odd consecutive integers.