Step-by-step explanation:
![\green{\large\underline{\sf{Solution-}}}](https://tex.z-dn.net/?f=%20%5Cgreen%7B%5Clarge%5Cunderline%7B%5Csf%7BSolution-%7D%7D%7D)
Given quadratic equation is
![\rm :\longmapsto\:\rm \: {(3x)}^{2} + \bigg(27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15 \bigg)x + 4 = 0](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%5Crm%20%5C%3A%20%20%7B%283x%29%7D%5E%7B2%7D%20%2B%20%5Cbigg%2827%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%20%20%5Cbigg%29x%20%2B%204%20%3D%200%20)
can be rewritten as
![\rm :\longmapsto\:\rm \: {9x}^{2} + \bigg(27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15 \bigg)x + 4 = 0](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%5Crm%20%5C%3A%20%20%7B9x%7D%5E%7B2%7D%20%2B%20%5Cbigg%2827%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%20%20%5Cbigg%29x%20%2B%204%20%3D%200%20)
<u>Concept Used :- </u>
Nature of roots :-
Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.
If Discriminant, D > 0, then roots of the equation are real and unequal.
If Discriminant, D = 0, then roots of the equation are real and equal.
If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.
Where,
Discriminant, D = b² - 4ac
Let's Solve this problem now!!!
On comparing with quadratic equation ax² + bx + c = 0, we get
![\red{\rm :\longmapsto\:a = 9}](https://tex.z-dn.net/?f=%5Cred%7B%5Crm%20%3A%5Clongmapsto%5C%3Aa%20%3D%209%7D)
![\red{\rm :\longmapsto\:b = 27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15}](https://tex.z-dn.net/?f=%5Cred%7B%5Crm%20%3A%5Clongmapsto%5C%3Ab%20%3D%2027%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%7D)
![\red{\rm :\longmapsto\:c = 4}](https://tex.z-dn.net/?f=%5Cred%7B%5Crm%20%3A%5Clongmapsto%5C%3Ac%20%3D%204%7D)
Since, Discriminant, D = 0
![\rm \implies\: {b}^{2} - 4ac = 0](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A%20%7Bb%7D%5E%7B2%7D%20-%204ac%20%3D%200)
![\rm :\longmapsto\: {\bigg(27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15\bigg)}^{2} - 4 \times 4 \times 9 = 0](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%20%7B%5Cbigg%2827%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%5Cbigg%29%7D%5E%7B2%7D%20%20-%204%20%5Ctimes%204%20%5Ctimes%209%20%3D%200)
![\rm :\longmapsto\: {\bigg(27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15\bigg)}^{2} - 144 = 0](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%20%7B%5Cbigg%2827%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%5Cbigg%29%7D%5E%7B2%7D%20%20-%20144%20%3D%200)
![\rm :\longmapsto\: {\bigg(27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15\bigg)}^{2} = 144](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%20%7B%5Cbigg%2827%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%5Cbigg%29%7D%5E%7B2%7D%20%20%3D%20144)
![\rm :\longmapsto\: {\bigg(27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15\bigg)}^{2} = {12}^{2}](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%20%7B%5Cbigg%2827%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%5Cbigg%29%7D%5E%7B2%7D%20%20%3D%20%20%7B12%7D%5E%7B2%7D%20)
![\rm \implies\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15 = \: \pm \: 12](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%20%3D%20%20%5C%3A%20%20%5Cpm%20%5C%3A%2012)
<u>Case - 1</u>
![\rm :\longmapsto\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15 = - 12](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%20%3D%20-%20%2012)
![\rm :\longmapsto\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } = - 12 + 15](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%20-%20%2012%20%2B%2015)
![\rm :\longmapsto\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } = 3](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%203)
![\rm \implies\:{\bigg[3\bigg]}^{ \dfrac{1}{k} } = \dfrac{1}{9}](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%20%5Cdfrac%7B1%7D%7B9%7D%20)
![\rm \implies\:{\bigg[3\bigg]}^{ \dfrac{1}{k} } = \dfrac{1}{ {3}^{2} }](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%20%5Cdfrac%7B1%7D%7B%20%7B3%7D%5E%7B2%7D%20%7D%20)
![\rm \implies\:{\bigg[3\bigg]}^{ \dfrac{1}{k} } = {3}^{ - 2}](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%20%20%7B3%7D%5E%7B%20-%202%7D%20%20)
![\rm \implies\:\dfrac{1}{k} = - 2](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A%5Cdfrac%7B1%7D%7Bk%7D%20%20%3D%20%20-%202)
![\bf\implies \:k \: = \: - \: \dfrac{1}{2}](https://tex.z-dn.net/?f=%5Cbf%5Cimplies%20%5C%3Ak%20%5C%3A%20%20%3D%20%20%5C%3A%20%20-%20%20%5C%3A%20%5Cdfrac%7B1%7D%7B2%7D%20)
<em>So, option (b) is Correct. </em>
<u>Case - 2</u>
![\rm :\longmapsto\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } - 15 = 12](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20-%2015%20%3D%2012)
![\rm :\longmapsto\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } = 12 + 15](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%2012%20%2B%2015)
![\rm :\longmapsto\:27 \times {\bigg[3\bigg]}^{ \dfrac{1}{k} } = 27](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A27%20%5Ctimes%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%2027)
![\rm :\longmapsto\: {\bigg[3\bigg]}^{ \dfrac{1}{k} } = 1](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%201)
![\rm :\longmapsto\: {\bigg[3\bigg]}^{ \dfrac{1}{k} } = {3}^{0}](https://tex.z-dn.net/?f=%5Crm%20%3A%5Clongmapsto%5C%3A%20%20%7B%5Cbigg%5B3%5Cbigg%5D%7D%5E%7B%20%5Cdfrac%7B1%7D%7Bk%7D%20%7D%20%3D%20%20%7B3%7D%5E%7B0%7D%20)
![\rm \implies\:\dfrac{1}{k} =0](https://tex.z-dn.net/?f=%5Crm%20%5Cimplies%5C%3A%5Cdfrac%7B1%7D%7Bk%7D%20%20%3D0)
<em>which is not possible.</em>