Question

If one zero of the polynomial $p(x) = (k + 4){x^2} + 13x + 13k$ is reciprocal of the other, then k is equal to
A) $2$
B) $3$
C) $4$
D) $5$

Answer 1

Hint: According to the question we have to determine the value of k if one zero of the polynomial $p(x) = (k + 4){x^2} + 13x + 13k$ is reciprocal of the other. So, first of all we have to let that one of the zero $t$ and the other zero is $\dfrac{1}{t}$
Now, we have to find the product of both of the roots.
Now, we have to determine the product of the roots of the general quadratic expression as $a{x^2} + bx + c = 0$which is as given below:
$ \Rightarrow $Product of the roots $ = \dfrac{c}{a}$………………………(A)
Now, we have to substitute the product in the expression and after some simple calculation we will obtain the value of the k.

Complete step-by-step solution:
Step 1: First of all we have to let the first zero and another zero which is reciprocal of the first zero. Hence,
First zero = $t$
Second zero = $\dfrac{1}{t}$
Step 2: Now, we have to multiply both of the roots as mentioned in the solution hint,
Product of the zeroes
$ \Rightarrow t \times \dfrac{1}{t} = 1$
Step 2: Now, we have to find the product of the roots for the given expression with the help of the formula (A) as mentioned in the solution hint. Hence, on comparing all the terms of the given expression with the help of the general expression,
$a = k + 4,b = 13,c = 3k$
Product of roots:
$ \Rightarrow \dfrac{c}{a} = \dfrac{{3k}}{{k + 4}}$
Step 3: Now, we have to substitute the product of the zeroes as we have obtained in step 1 in the expression as obtained in the solution step 2. Hence,
$ \Rightarrow \dfrac{{3k}}{{k + 4}} = 1$
On applying cross-multiplication in the expression obtained just above,
$
   \Rightarrow 3k = k + 4 \\
   \Rightarrow 3k - k = 4 \\
   \Rightarrow 2k = 4 \\
   \Rightarrow k = \dfrac{4}{2} \\
   \Rightarrow k = 2
 $
Hence, with the help of formula (A) as mentioned in the solution hint we have obtained the value$k = 2$.

Therefore correct option is (A)

Note: In the quadratic expression there can be only two zeroes and the general form of quadratic expression is $a{x^2} + bx + c = 0$ and the product of the zeroes for this given expression is $\dfrac{c}{a}$
To obtain the value of a product of the given expression it is necessary to compare the given expression with the general form of expression.