Question

if the sum of the zero of the polynomial p(x)=(k2-14)x2-2x-12 is 1,then find the value of k​

Answer 1

Answer :

The value of k is 4

Given :

The quadratic polynomial is :

• p(x) = (k² - 14)x² - 2x - 12
• Sum of the zeroes of the polynomial is 1

To Find :

• The value of k

Concept to be used :

Relationship between the zeroes and the coefficients of the polynomial :

$\sf \bullet \: \: Sum \: of \: the \: zeroes = -\dfrac{Coefficient \: of \: x}{Coefficient \: of \: x^{2}} \\\\ \sf \bullet \: \: Product\: the \: zeroes = \dfrac{Constant \: term}{Coefficient \: of \: x^{2}}$

Solution :

Given , the sum of the zeroes of the given polynomial is 1

Applying the relation of sum of zeroes

$\sf \dashrightarrow Sum\: of \: the \: zeroes = -\dfrac{Coefficient\: of \:x}{Coefficient \: of \: x^{2}} \\\\ \sf \implies 1 = -\dfrac{-2}{k^{2} - 14}\\\\ \sf \implies k^{2} - 14 = 2 \\\\ \sf \implies k^{2} = 14+2 \\\\ \sf \implies k^{2} = 16 \\\\ \sf \implies k^{2} = 4^{2} \\\\ \sf \implies k= 4$

Thus , the required value of k is 4

Answer 2

Given ,

i) The polynomial is

(k² - 14)x² - 2x - 12

ii) The sum of zeroes of the polynomial is 1

We know that ,

$\sf \star \: \: \fbox{ \mathtt{Sum \: of \: zeroes = - \frac{b}{a} }}$

Thus ,

$\Rightarrow \bold{1 = - \frac{ (- 2)}{ {(k)}^{2} - 14 }} \\ \\ \Rightarrow \bold{{(k)}^{2} - 14 = 2} \\ \\ \Rightarrow \bold{ {(k)}^{2} = 16} \\ \\\Rightarrow \bold{ k = \sqrt{16} } \\ \\ \Rightarrow \bold{k = 4}</p><p>$

$\therefore \sf \underline{ \bold{ The \: value \: of \: k \: is \: 4 \:}}$