Question

if a, B are the zeroes of the quadratic polynomial x^2+3x+6, find the value of (a/B+B/a).​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: \alpha \: and \: \beta \: are \: zeroes \: of \: {x}^{2} + 3x + 6$

We know,

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

$\rm :\implies\: \alpha + \beta = - \: \dfrac{3}{1} = - \: 3 - - - (1)$

And

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

$\rm :\implies\: \alpha \beta = \dfrac{6}{1} = 6 - - - (2)$

Now, we have to find the value of

$\rm :\longmapsto\:\dfrac{ \alpha }{ \beta } + \dfrac{ \beta }{ \alpha }$

$\: \rm \: = \:\dfrac{ { \alpha }^{2} + { \beta }^{2} }{ \alpha \beta }$

$\rm \: = \: \: \dfrac{ {( \alpha + \beta )^{2} - 2 \alpha \beta }}{6} \: \: \: \: \: \: \: \: \: \{ \: using \: (2) \}$

$\: \rm \: = \:\dfrac{ {( - 3)}^{2} - 2 \times 6 }{6}$

$\: \rm \: = \:\dfrac{9 - 12}{6}$

$\: \rm \: = \:\dfrac{ - 3}{6}$

$\: \rm \: = - \:\dfrac{1}{2}$

Hence,

$\bf\implies \:\:\dfrac{ \alpha }{ \beta } + \dfrac{ \beta }{ \alpha } = - \: \dfrac{1}{2}$

Additional Information :-

$\rm :\longmapsto\: \alpha, \beta, \gamma \: are \: zeroes \: of \: {ax}^{3} + {bx}^{2} + cx+ d \: then$

$\boxed{ \sf{ \: \alpha + \beta + \gamma = - \: \frac{b}{a}}}$

$\boxed{ \sf{ \: \alpha \beta + \beta \gamma + \gamma \alpha = \: \frac{c}{a}}}$

$\boxed{ \sf{ \: \alpha\beta \gamma = - \: \frac{d}{a}}}$

Answer 2

${\huge{\boxed{\tt{\color {red}{A᭄ɴsᴡᴇʀ࿐}}}}}$

6x^× +x—2 =0

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =0

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/2

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2x= --2/3

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2x= --2/3a = --2/3. b. . =1/2

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2x= --2/3a = --2/3. b. . =1/2Find the value of a/b +b/a

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2x= --2/3a = --2/3. b. . =1/2Find the value of a/b +b/aa/b = -- 4/3 b/ a = --3/4

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2x= --2/3a = --2/3. b. . =1/2Find the value of a/b +b/aa/b = -- 4/3 b/ a = --3/4a/b+b/a = -- 4/3 -- 3/4 = (—16 --9) /12 = --25 /12

6x^× +x—2 =0a and b are zeroes of QUADRATIC EQUATION6x^2 +4x--3x -2 =0 ( 6× -2 =--12 4×--3= --12 +4--3 =1)2x(3x +2 ) -- 1(3x+2) =0(2x-1 ) (3x+-2) =02x=1. x=1/23x=--2x= --2/3a = --2/3. b. . =1/2Find the value of a/b +b/aa/b = -- 4/3 b/ a = --3/4a/b+b/a = -- 4/3 -- 3/4 = (—16 --9) /12 = --25 /12The required value is --25/12