Question 1209123
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Vieta's Theorem tells us that, if a and b are the roots of x^2+7x-4=0, then...<br>
sum of roots = a+b = -7
product of roots = ab = -4<br>
Manipulate the given expression {{{(a+3)/(b+3)+(b+3)/(a+3)}}} to write it entirely in terms of (a+b), (ab), and constants.<br>
{{{(a+3)/(b+3)+(b+3)/(a+3)}}}<br>
{{{((a+3)^2+(b+3)^2)/((a+3)(b+3))}}}<br>
{{{(a^2+6a+9+b^2+6b+9)/(ab+3a+3b+9)}}}<br>
{{{(a^2+2ab+b^2-2ab+6a+6b+18)/(ab+3a+3b+9)}}}<br>
{{{((a+b)^2-2(ab)+6(a+b)+18)/((ab)+3(a+b)+9)}}}<br>
{{{((-7)^2-2(-4)+6(-7)+18)/((-4)+3(-7)+9)}}}<br>
{{{(49+8-42+18)/(-4-21+9)}}}<br>
{{{(33)/(-16)}}}<br>
ANSWER: -33/16<br>
Note this solution makes use of an identity that is useful in solving many algebraic problems: {{{a^2+b^2=(a+b)^2-2ab}}}<br>