Question

If ${x^2} - 2x - 15 = (x + r)(x + s)$ for all $x \in R$, then what is/are the possible value(s) of $r - s$?
A. $ - 3$
B. $ - 2$
C. $2$
D. $8$

Answer 1

Hint: In this question, we are given an equation and we are asked to find the difference between two variables given in the equation. To find the difference, we are first required to find the possible values of the two given variables. We will start by splitting the middle term of the LHS of the equation so that we can bring it in the same format as of the RHS. Then, we will compare both the sides and find the possible values.

Complete step-by-step answer:
We are given an equation ${x^2} - 2x - 15 = (x + r)(x + s)$. We will begin by splitting the middle term of the LHS of the given equation:
$ \Rightarrow {x^2} - 2x - 15$
Find two factors such that, when they are added they give $ - 2$ and when they are multiplied, they give $ - 15$. From observation, such factors are $ - 5$ and $3$.
Hence, splitting the middle term of the given equation.
$ \Rightarrow {x^2} - 5x + 2x - 15$
Taking common and grouping,
$ \Rightarrow x\left( {x - 5} \right) + 3(x - 5)$
$ \Rightarrow (x + 3)(x - 5)$
Now, if we compare the LHS with the RHS, we will get two values of $r$ and $s$.
$ \Rightarrow (x + 3)(x - 5) = (x + r)(x + s)$
On comparing we get,
$r = 3,s = - 5$ or $s = 3,r = - 5$
Now, subtracting both to find $r - s$.
$ \Rightarrow r - s = 3 - ( - 5)$
Simplifying,
$ \Rightarrow r - s = 3 + 5$
$ \Rightarrow r - s = 8$
Taking the other values,
$ \Rightarrow r - s = - 5 - 3$
Simplifying,
$ \Rightarrow r - s = - 8$
Hence, we get the two values of $r - s$.
$ \Rightarrow r - s = 8, - 8$
But, we have only $8$ as an option in the given options.
Therefore, the value $r - s = - 8$ will be discarded and $r - s = 8$ will be chosen.

Option D is the correct answer

Note: Even though the options do not have $ - 8$ as an answer, we will still show the complete solution as $ - 8$ is one of the right answers of the given question and we can only judge the correct option if we have all the possible answers. And also students must focus on the calculations of factorization of the given equation.