Notes: 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...

Thursday, 29 January 2015

Q. The value of $\displaystyle{a}$ for which the sum of the squares of the roots of the equation $\displaystyle{{x}}^{{2}}-{\left({a}-{2}\right)}{x}-{a}-{1}={0}$ assume the least value is:- A)2 B)3 C)0...

We are given
the equation $\displaystyle{{x}}^{{2}}-{\left({a}-{2}\right)}{x}-{a}-{1}={0}$ 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):

$\displaystyle{r}_{{1}},{r}_{{2}}=\frac{{{a}-{2}\pm\sqrt{{{{\left({a}-{2}\right)}}^{{2}}-{4}{\left({1}\right)}{\left(-{\left({a}+{1}\right)}\right)}}}}}{{2}}$

$\displaystyle=\frac{{{a}-{2}\pm\sqrt{{{{a}}^{{2}}-{4}{a}+{4}+{4}{a}+{4}}}}}{{2}}$

$\displaystyle=\frac{{{a}-{2}\pm\sqrt{{{{a}}^{{2}}+{8}}}}}{{2}}$

$\displaystyle=\frac{{{a}-{2}}}{{2}}\pm\frac{\sqrt{{{{a}}^{{2}}+{8}}}}{{2}}$

So $\displaystyle{{r}_{{1}}^{{2}}}+{{r}_{{2}}^{{2}}}=$

$\displaystyle{{\left(\frac{{{a}-{2}}}{{2}}+\frac{\sqrt{{{{a}}^{{2}}+{8}}}}{{2}}\right)}}^{{2}}+{{\left(\frac{{{a}-{2}}}{{2}}-\frac{\sqrt{{{{a}}^{{2}}+{8}}}}{{2}}\right)}}^{{2}}$

$\displaystyle=\frac{{{{\left({a}-{2}\right)}}^{{2}}}}{{4}}+{2}\frac{{{a}-{2}}}{{2}}\frac{\sqrt{{{{a}}^{{2}}+{8}}}}{{2}}+\frac{{{{a}}^{{2}}+{8}}}{{4}}$

$\displaystyle+\frac{{{\left({a}-{2}\right)}}^{{2}}}{{4}}-{2}\frac{{{a}-{2}}}{{2}}\frac{\sqrt{{{{a}}^{{2}}+{8}}}}{{2}}+\frac{{{{a}}^{{2}}+{8}}}{{4}}$

$\displaystyle=\frac{{{\left({a}-{2}\right)}}^{{2}}}{{2}}+\frac{{{{a}}^{{2}}+{8}}}{{2}}$

$\displaystyle=\frac{{{{a}}^{{2}}-{4}{a}+{4}+{{a}}^{{2}}+{8}}}{{2}}$

$\displaystyle=\frac{{{2}{{a}}^{{2}}-{4}{a}+{12}}}{{2}}$

$\displaystyle={{a}}^{{2}}-{2}{a}+{6}$

This is minimized
at a=1 (The graph of $\displaystyle{y}={{a}}^{{2}}-{2}{a}+{6}$ 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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Notes

Thursday, 29 January 2015

Q. The value of $\displaystyle{a}$ for which the sum of the squares of the roots of the equation $\displaystyle{{x}}^{{2}}-{\left({a}-{2}\right)}{x}-{a}-{1}={0}$ assume the least value is:- A)2 B)3 C)0...

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In 1984, is Julia a spy? Please provide specific examples from the book. My teacher says that he knows of 17 pieces of evidence which proves that Julia...

Thursday, 29 January 2015

Q. The value of for which the sum of the squares of the roots of the equation assume the least value is:- A)2 B)3 C)0...

No comments:

Post a Comment

In 1984, is Julia a spy? Please provide specific examples from the book. My teacher says that he knows of 17 pieces of evidence which proves that Julia...

Q. The value of $\displaystyle{a}$ for which the sum of the squares of the roots of the equation $\displaystyle{{x}}^{{2}}-{\left({a}-{2}\right)}{x}-{a}-{1}={0}$ assume the least value is:- A)2 B)3 C)0...