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If α And β Are the Zeros of the Quadratic Polynomial P(S) = 3s2 − 6s + 4, Find the Value of `Alpha/Beta+Beta/Alpha+2[1/Alpha+1/Beta]+3alphabeta` - CBSE Class 10 - Mathematics

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Question

If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4, find the value of `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta`

Solution

Since α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4

`alpha+beta="-coefficient of x"/("coefficient of "x^2)`

`alpha+beta=(-(-6))/3`

`alpha+beta=6/3`

`alpha+beta=2`

`alphabeta="constant term"/("coefficient of "x^2)`

`alphabeta=4/3`

We have, `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta`

`=(alpha^2+beta^2)/(alphabeta)+2[1/alpha+1/beta+3alphabeta]`

`=((alpha+beta)^2-2alphabeta)/(alphabeta)+2[(alpha+beta)/(alphabeta)]+3alphabeta`

By substituting `alpha+beta=2 " and "alphabeta=4/3` we get,

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=((2)^2-2(4/3))/(4/3)+2((2))/(4/3)+3(4/3)`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=(4-8/3)/(4/3)+4/(4/3)+12/3`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=((4xx3)/(1xx3)-8/3)/(4/3)+4/(4/3)+12/3`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=((12-8)/3)/(4/3)+4/(4/3)+12/3`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=(4/3)/(4/3)+4/(4/3)+12/3`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=4/3xx3/4+(4xx3)/4+12/3`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=1+12/4+12/3`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=(1xx12)/(1xx12)+(12xx3)/(4xx3)+(12xx4)/(3xx4)`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=(12+36+48)/12`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=(48+48)/12`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=96/12`

`alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta=8`

Hence, the value of `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta " is "8`

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APPEARS IN

 RD Sharma Solution for 10 Mathematics (2018 to Current)
Chapter 2: Polynomials
Ex. 2.10 | Q: 10 | Page no. 34
Solution If α And β Are the Zeros of the Quadratic Polynomial P(S) = 3s2 − 6s + 4, Find the Value of `Alpha/Beta+Beta/Alpha+2[1/Alpha+1/Beta]+3alphabeta` Concept: Relationship Between Zeroes and Coefficients of a Polynomial.
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