Question

How do you find two positive consecutive odd integers whose product is 483?

Answer 1

Hint: Assume the two positive consecutive odd integers as (x) and (x + 2). Take their product and equate with 483. Form a quadratic equation in x and use the middle term split method to get the values of x. neglect the negative value of x and select the positive value obtained. Substitute this value in the two odd integers assumed to get the answer.

Complete step by step answer:
Here we have been given that the product of two positive consecutive odd integers is 483 and we are asked to determine such two integers.
Now, let us select the first odd positive integer as x so the next integer will be (x + 1) but this will be an even integer therefore we cannot select it. Now, the next integer will be (x + 2) which will be an odd integer, so the two consecutive odd positive integers can be selected as x and (x + 2). Taking the product of these two integers and equating with 483, we get,
$\begin{align}
  & \Rightarrow x\left( x+2 \right)=483 \\
 & \Rightarrow {{x}^{2}}+2x=483 \\
 & \Rightarrow {{x}^{2}}+2x-483=0 \\
\end{align}$
Clearly we can see that the above equation is a quadratic equation, so let us apply the middle term split method to solve this equation. Splitting the middle term into terms such that their sum is 2x and the product is $-483{{x}^{2}}$, we get,
$\begin{align}
  & \Rightarrow {{x}^{2}}+23x+\left( -21x \right)-483=0 \\
 & \Rightarrow {{x}^{2}}+23x-21x-483=0 \\
 & \Rightarrow x\left( x+23 \right)-21\left( x+23 \right)=0 \\
 & \Rightarrow \left( x-21 \right)\left( x+23 \right)=0 \\
\end{align}$
Substituting each term equal to 0 we get,
\[\Rightarrow x=21\] or $x=-23$
Since we need to consider only the positive odd integer so we need to neglect x = -23. So we have,
$\therefore x=21$ and $\left( x+2 \right)=23$

Hence, the two consecutive odd positive integers are 21 and 23.

Note: You must be careful about selecting the consecutive odd integers or you may get the wrong quadratic expression and answer. You can also assume the integers as (x + 1) and (x + 3) and there can be infinite such assumptions. After finding the values of x always check for the invalid value according to the conditions given in the question.