Question

If one zero of 7x²+ 6x + k is the reciprocal of the other,find the value of k.​

Answer 1

$\begin{gathered}\begin{gathered}\bf \: Given \: - \begin{cases} &\sf{a \: polynomial \: {7x}^{2} + 6x + k} \\ &\sf{having \: zeroes \: reciprocal \: of \: each \: other} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: find \: - \begin{cases} &\sf{value \: of \: k} \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

$\begin{gathered}\begin{gathered}\bf \: Let \: - \begin{cases} &\sf{one \: zero \: be \: \alpha } \\ &\sf{other \: zero \: be \: \dfrac{1}{ \alpha } } \end{cases}\end{gathered}\end{gathered}$

Now

We know

$\boxed{\purple{\tt Product\ of\ the\ zeroes=\frac{c}{a}}}$

OR

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

So,

$\rm \: \rightarrow \: \alpha \times \dfrac{1}{ \alpha } = \dfrac{k}{7}$

$\rm : \implies \:1 \: = \: \dfrac{k}{7}$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \:k \: = \: 7}}$

Remark :-

ShortCut :-

• If the roots of the polynomial f(x) = ax² + bx + c are reciprocal of each other, then a = c.

Now,

For the given statement,

• The roots of the polynomial 7x² + 6x + k are reciprocal of each other,

therefore,

• k = 7.

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Explore more

$\boxed{\purple{\tt Sum\ of\ the\ zeroes=\frac{-b}{a}}}$

OR

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \: { \alpha }^{2} + { \beta }^{2} = {( \alpha + \beta )}^{2} - 2 \alpha \beta }}$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \: { \alpha }^{3} + { \beta }^{3} = {( \alpha + \beta )}^{3} - 3 \alpha \beta( \alpha + \beta ) }}$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \: {( \alpha - \beta )}^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta }}$

Answer 2

Answer:

the value of p=7

Step by Step explanation:-

We know that, product of the zeroes is c/a

alpha×1/alpha=c/a

alpha and alpha gets cancelled out and we get the remaining product as,

1=c/a

here c/a=p/7

then,1=p/7

when we transpose 7 to the LHS(left hand side)we get the value of p to be 7

thus the value of p=7

hope it helps yu☺️