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Question Number 12543 by @ANTARES_VY last updated on 25/Apr/17

This

x^{2} - α x + α - 1 = 0 .

the roots of the equation x_{1} and x_{2}

a {what}^{'} s the value of x_{1}^{2} + x_{2}^{2} this collection of

smille (minimum) value .

Answered by mrW1 last updated on 25/Apr/17

x^{2} - α x + α - 1 = (x - x_{1}) (x - x_{2})

x^{2} - α x + α - 1 = x^{x} - (x_{1} + x_{2}) x + x_{1} x_{2}

x_{1} + x_{2} = α

x_{1} x_{2} = α - 1

x_{1}^{2} + x_{2}^{2} + 2 x_{1} x_{2} = α^{2}

x_{1}^{2} + x_{2}^{2} = α^{2} - 2 α + 2 = {(α - 1)}^{2} + 1 ⩾ 1

Commented by @ANTARES_VY last updated on 25/Apr/17

{(α - 1)}^{2} + 1 ⩾ 1 ? ? ? ? ?

Commented by @ANTARES_VY last updated on 25/Apr/17

I do not understand

Commented by mrW1 last updated on 25/Apr/17

s i n c e {(α - 1)}^{2} ⩾ 0

h e n c e {(α - 1)}^{2} + 1 ⩾ 1

Answered by ridwan balatif last updated on 25/Apr/17

x^{2} - α x + α - 1 = 0

x_{1} + x_{2} = α

x_{1} . x_{2} = α - 1

let p = x_{1}^{2} + x_{2}^{2}

p = {(x_{1} + x_{2})}^{2} - {2 x}_{1} x_{2}

p = {(α)}^{2} - 2 (α - 1)

p = α^{2} - 2 α + 2

p will minimum if \frac{dp}{d α} = 0

2 α - 2 = 0

α = 1

so the minimum value of

p = x_{1}^{2} + x_{2}^{2}

= α^{2} - 2 α + 2

= 1^{2} - 2.1 + 2

p = 1

OR

i think this is what MrW 1 think

p = α^{2} - 2 α + 2

p = {(α - 1)}^{2} - 1 + 2

p = {(α - 1)}^{2} + 1

p will minimum if {(α - 1)}^{2} = 0, so

p = 1