User:Egm6321.f12.team4.harris/Homework56 R*6.6 Part 2.5

Problem R*6.6 2.5 – Proof of Coefficients
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Statement
Show that $$ \alpha \beta = \frac{a_0}{a_2} $$ and $$ \alpha + \beta = \frac{a_1}{a_2} $$, thus making $$ \alpha $$ and $$ \beta $$ solutions of the characteristic equation
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$$ (\lambda - \alpha)(\lambda - \beta) = \lambda ^2 - (\alpha + \beta )\lambda + \alpha \beta $$     (6.6.?)
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Solution
Given the quadratic equation for $$ \alpha $$ from part 2.2 above as
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$$ a_2 \alpha ^2 - a_1 \alpha + a_0 = 0 $$     (6.6.?)
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one can divide the entire equation by $$ a_2 $$ to obtain a coefficient of $$ 1 $$ on the $$ \alpha ^2 $$ term. One can then equate this result to the right hand side of equation 6.6.? in the problem statement such that
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$$ \alpha ^2 - \frac{a_1}{a_2} \alpha + \frac{a_0}{a_2} = 0 = \lambda ^2 - (\alpha + \beta )\lambda + \alpha \beta $$     (6.6.?)
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By simple observation, one can see that, in the above equation, $$ \alpha + \beta = \frac{a_1}{a_2} $$ and $$ \alpha \beta = \frac{a_0}{a_2} $$.