Question

If $\alpha and \beta$ are the zeros of the quadratic polynomial $p(s)= 3s^{2}-6s+4$ ,find the value of $\frac{\alpha}{\beta} +\frac{\beta}{\alpha} +2 (\frac{1}{\alpha} +\frac{1}{\beta})+3 \alpha \beta$

Answer 1

SOLUTION :

Given :  α and β are the zeroes of the quadratic polynomial p(s)= 3s² - 6s + 4.

On comparing with as² + bs + c,

a = 3  , b= - 6  , c= 4

Sum of the zeroes = −coefficient of x / coefficient of x²

α + β  = -b/a = -(-6/3) = 2

α + β = 2  ……………………………(1)

Product of the zeroes = constant term/ Coefficient of x²

αβ = c/a = 4/3  

αβ =  4/3 ………………………….(2)

Now,

α/β + β/α +  2[1/α+1/β] + 3αβ.

= α²+β² /αβ + 2[(α+β)/αβ]+ 3αβ

As we know that ,  α + β)² = α² + β² +  2αβ ,  α² + β² = (α + β)² - 2αβ

= (α+β)² –2αβ / αβ + 2[(α+β)/αβ]+ 3αβ

By Substituting the value from eq 1 & 2

CALCULATION IS IN THE ATTACHMENT  

Hence, the value is α/β + β/α +  2[1/α+1/β] + 3αβ is 8.

HOPE THIS ANSWER WILL HELP YOU…