Question

if α and β are roots of the quadratic equation 3x^2+kx+8=0 and α/β =2/3, then find the value of k.

Jaldi batao nhi ti mia khalufa ki shaadi ho jayegi​

Answer 1

Step-by-step explanation:

Given that :

α and β are the roots of Equation

3x²+kx+8=0

and α/β = 2 / 3

We know that :-

a² + b² = (a + b)² - 2 ab

and

For any Quadratic equation:

ax² + bx + c = 0

if α and β are the roots of equation

=> α + β = -b/a , α×β = c / a

here,

α/β = 2/3 ....(1.)

we also say that

β/α = 3/2 .... (2.)

Adding 1 and 2

we get

α/β + β/α = 2/3 + 3/2

=> (α² + β²) / αβ = (4 + 9) / 6

=> (α² + β²) / αβ = 13 / 6 ...... (3.)

Now,

From given quadratic equation

3x²+kx+8=0

{ a = 3 , b = k , c = 8 }

here,

α+ β = -k / 3 & α×β = 8 /3

From here,

α² + β² = (α + β)² - 2 αβ

=> ( -k/ 3)² - 2 × ( 8 / 3 )

=>( k² / 9 ) - (16 / 3 )

=> α² + β² = ( k² - 48) / 9

putting value in ...3

=> [ { ( k² - 48) / 9 } / ( 8 /3) ] = 13 / 6

=> ( k² - 48) / 24 = 13 / 6

=> ( k² - 48) = 52

=> k² = 52 - 48 = 4

=> k = ± 2 Answer

Hope it Helps...

:-)

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