Question

The product of two consecutive even integers is $624$.How do you find the integers?

Answer 1

Hint:In order the determine the two consecutive even integers ,let them be $\left( x \right){\kern 1pt} \,and\,\left( {x + 2} \right)$ and put into a mathematical expression$\left( x \right)\left( {x + 2} \right) = 624$. You will get a quadratic equation whose solution can be obtained using the splitting middle term method by finding the factors of $64$as $24 \times 26$ to obtain your desired solution.

Complete step by step solution:
We are given that the product of two consecutive even integers is $624$.

Consecutive numbers are numbers which come just after the previous in the continuous order.

Let the two consecutives even integers be $\left( x \right){\kern 1pt} \,and\,\left( {x + 2} \right)$.

Let’s frame the statement into mathematical form

Product of Two consecutive even number=$624$

$
\Rightarrow \left( x \right)\left( {x + 2} \right) = 624 \\
\Rightarrow {x^2} + 2x = 624 \\
\Rightarrow {x^2} + 2x - 624 = 0 \\
$

In order to solve the quadratic equation use the splitting up the middle term method.

Middle term can be split into $26x - 24x$as the multiplication of these number $26 \times 24 = 624$

$
\Rightarrow {x^2} + 2x - 168 = 0 \\
\Rightarrow {x^2} + 26x - 24x - 168 = 0 \\
$

Now taking common $x$from the first two terms and \[ - 24\]from the last two terms.

$
\Rightarrow x\left( {x + 26} \right) - 24\left( {x + 26} \right) = 0 \\
\Rightarrow \left( {x + 26} \right)\left( {x - 24} \right) = 0 \\
$

So value of $x = - 26,24$
When $x = 24$

Then the consecutive even integers become $24,26$
And when $x = - 26$ the consecutive even integers are $ - 24, - 26$

Therefore there are two possible solution for the value of consecutive even integers
1. $24,26$(when$x = 24$)
2. $ - 24, - 26$(when $x = - 26$)

Alternate:
You can also alternatively use a direct method which uses Quadratic Formula to find both roots of a quadratic equation as

$x1 = \dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}$ and $x2 = \dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}$
x1,x2 are root to quadratic equation $a{x^2} + bx + c$

Hence the factors will be $(x - x1)\,and\,(x - x2)\,$.

Additional Information: 1.Mathematical equation : A Mathematical equation can be defined as the mathematical statement which contains an equal symbol $ = $ in between two algebraic expressions that share the same value.

2.$2x + 98 + 78y$is not a mathematical equation because it does not contain equality $ = $ symbol . It is only a mathematical expression.

3. Quadratic Equation: A quadratic equation is a equation which can be represented in the form of $a{x^2} + bx + c$where $x$is the unknown variable and a,b,c are the numbers known where $a \ne 0$.If $a = 0$then the equation will become a linear equation and will no longer be quadratic .

Note:1. Read the statement carefully in order to convert them into mathematical expressions.

2.Don’t forget to cross-check your answer at least once.