If - and - are the roots of the quadratic equation x2-p-3x-2p-3 p- R- then the minimum value of -2-2- is

The correct option is C 8 x ^ 2 +(p 3) 2p=3 if α #160;and β #160;are the roots of this equation, #160;then α+β=p 3 1 and ...

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Standard XII

Mathematics

Algebra of Limits

If α and ...

Question

If $α$ and $β$ are the roots of the quadratic equation $x^{2} + (p - 3) x - 2 p = 3$ $(p \in R)$ , then the minimum value of $(α^{2} + β^{2} + α β)$ , is

2

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Solution

The correct option is C $8$
$x^{2} + (p - 3) - 2 p = 3$ if $α$ and $β$ are the roots of this equation, then $α + β = \frac{p - 3}{- 1}$ and $α β = \frac{- 2 p - 3}{1}$ .
Now, $α^{2} + β^{2} + α β = {(α + β)}^{2} - α β$ .
$= {(3 - p)}^{2} + (2 p + 3)$
$= p^{2} - 4 p + 12$
$= (p - 2)^{2} + 8$
Therefore, minimum value is obtained at $p = 2$ .
Minimum value $= 8$ .

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If α and β are the roots of the quadratic equation x2+(p−3)x−2p=3 (p∈R), then the minimum value of (α2+β2+αβ), is

The correct option is C 8x2+(p−3)−2p=3 if α and β are the roots of this equation, then α+β=p−3−1 and αβ=−2p−31.Now,α2+β2+αβ=(α+β)2−αβ.=(3−p)2+(2p+3)=p2−4p+12=(p−2)2+8Therefore, minimum value is obtained at p=2.Minimum value=8.

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