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Q. The value of `a` for which the sum of the squares of the roots of the equation...

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user8235304 | Student, Grade 11 | Valedictorian

Posted August 12, 2013 at 12:41 AM via web

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Q. The value of `a` for which the sum of the squares of the roots of the equation `x^2-(a-2)x-a-1=0` assume the least value is:-

A)2

B)3

C)0

D)1

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted August 12, 2013 at 2:10 AM (Answer #1)

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We are given the equation `x^2-(a-2)x-a-1=0` and we are asked to find the value of a that minimizes the sum of the square of the roots:

We use the quadratic formula to find the roots with a=1,b=-(a-2), and c=-(a+1):

`r_1,r_2=(a-2+-sqrt((a-2)^2-4(1)(-(a+1))))/2`

`=(a-2+-sqrt(a^2-4a+4+4a+4))/2`

`=(a-2+-sqrt(a^2+8))/2`

`=(a-2)/2 +- sqrt(a^2+8)/2`

So `r_1^2+r_2^2=`

`((a-2)/2+sqrt(a^2+8)/2)^2+((a-2)/2-sqrt(a^2+8)/2)^2`

`=((a-2)^2)/4+2(a-2)/2 sqrt(a^2+8)/2+(a^2+8)/4`

`+(a-2)^2/4-2(a-2)/2 sqrt(a^2+8)/2+(a^2+8)/4`

`=(a-2)^2/2+(a^2+8)/2`

`=(a^2-4a+4+a^2+8)/2`

`=(2a^2-4a+12)/2`

`=a^2-2a+6`

This is minimized at a=1 (The graph of `y=a^2-2a+6` is a parabola opening up; the minimum occurs at the vertex.)

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The minimum of the sum of the squares of the roots occurs when a=1

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