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The value of $b$ for which the sum of the squares of the roots of the equation $x^{2}-(b-2)x-b-1=0$ assumes the least value is

Option 1)
0

Option 2)
1

Option 3)
2

Option 4)
3

Answers (1)

best_answer

As learnt in concept

Roots of Quadratic Equation with real Coefficients -

$\alpha ,\beta$ are roots if

$ax^{2}+bx+c= 0$

is satisfied by $x= \alpha ,\beta$

- wherein

$\alpha ,\beta\in C$

$a,b,c\in R$

And,

Sum of Roots in Quadratic Equation -

$\alpha +\beta = \frac{-b}{a}$

- wherein

$\alpha \: and\beta$ are root of quadratic equation

$ax^{2}+bx+c=0$

$a,b,c\in C$

And,

Product of Roots in Quadratic Equation -

$\alpha \beta = \frac{c}{a}$

- wherein

$\alpha \: and\ \beta$ are roots of quadratic equation:

$ax^{2}+bx+c=0$

$a,b,c\in C$

$\alpha ^{2}+ \beta ^{2}=(\alpha + \beta)^{2}-2 \alpha \beta$

$=(b-2)^{2}-2(-1)(b+1)$

$=b^2+4-4b+2b+2$

$=b^{2}-2b+6$

$=b^{2}-2b+1+5$

$=(b-1)^{2}+5$

Minimum value at b=1

Option 1)

0

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Option 2)

1

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Option 3)

2

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Option 4)

3

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