Question 1209715
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The AI solution from the other tutor is wrong from the first line, since it uses roots 1/2 and 7 instead of 12 and 7.<br>
The process used, however, is fine.<br>
Follow the process shown in that solution using the correct two given roots to find....<br>
third root: -191/38<br>
a = -531/19<br>
c = 16044/19<br>
Then you can check those results by graphing the polynomial using desmos.com<br>
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NOTE: it is possible that the original problem somehow got corrupted, and that the correct give roots were in fact 1/2 and 7.  I say that because those roots give much "nicer" answers for the third root (and for the coefficients a and c).<br>
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While that path to the answer is fine, another path is perhaps easier, requiring arithmetic that is not quite as nasty.<bt>
{{{P(x) = 2x^3 + ax^2 - 23x + c}}}<br>
7 and 12 are roots:<br>
7: {{{2(7^3)+a(7^2)-23(7)+c=0}}}
{{{686+49a-161+c=0}}}
[1] {{{49a+c=-525}}}<br>
12: {{{2(12^3)+a(12^2)-23(12)+c=0}}}
{{{3456+144a-276+c=0}}}
[2] {{{144a+c=-3180}}}<br>
Comparing [1] and [2]...<br>
{{{95a=-2655}}}
{{{a=-2655/95=-531/19}}}<br>
Then<br>
{{{49(-531/19)+c=-525}}}<br>
{{{c=-525+49*531/19=16044/19}}}<br>
So (a,c) = (-531/19,16044/19)<br>
Find the third root knowing that the sum of the roots is -a/2.<br>
{{{12+7+r=531/38}}}
{{{r=531/38-19=(531-722)/38=-191/38}}}<br>