Question

The product of two consecutive even integers is $ 168 $ .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. You will get a quadratic equation whose solution can be obtained using the splitting middle term method or quadratic formula.

Complete step-by-step answer:
We are given that the product of two consecutive even integers is $ 168 $ .
Consecutive number are numbers which come just after the previous in the continuous order
Let the two consecutive even integers be $ \left( x \right){\kern 1pt} \,and\,\left( {x + 2} \right) $ .
Lets frame the statement into mathematical form
Product of Two consecutive even number= $ 168 $
 $
   \Rightarrow \left( x \right)\left( {x + 2} \right) = 168 \\
   \Rightarrow {x^2} + 2x = 168 \\
   \Rightarrow {x^2} + 2x - 168 = 0 \;
 $
In order to solve the quadratic equation use the splitting up the middle term method.
Middle term can be split into $ 14x - 12x $ as the multiplication of these number $ 12 \times 14 = 168 $
 $
   \Rightarrow {x^2} + 2x - 168 = 0 \\
   \Rightarrow {x^2} + 14x - 12x - 168 = 0 \;
 $
Now taking common $ x $ from the first two terms and \[ - 12\]from the last two terms.
 $
   \Rightarrow x\left( {x + 14} \right) - 12\left( {x + 14} \right) = 0 \\
   \Rightarrow \left( {x + 14} \right)\left( {x - 12} \right) = 0 \;
 $
So value of $ x = 12, - 14 $
When $ x = 12 $
Then the consecutive even integers become $ 12,14 $
And when $ x = - 14 $
the consecutive even integers are $ - 14, - 12 $
Therefore there are two possible solution for the value of consecutive even integers
1. $ 12,14 $
2. $ - 14, - 12 $
So, the correct answer is “1. $ 12,14 $ OR 2. $ - 14, - 12 $ ”.

Note: 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 linear equation and will no more quadratic