Question 1209228
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Answer: <font color=red>569/16</font>
569/16 = 35.5625 exactly without any rounding done to it.


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Explanation


2x^2 - 8x + 7 = -3x^2 + 15x + 11 
rearranges to
5x^2 - 23x - 4 = 0
after getting everything to one side.


Divide everything by the leading coefficient
x^2 - (23/5)x - 4/5 = 0
This is to make the leading coefficient be equal to 1.


Vieta's Formulas say that the roots add to the negative of the x coefficient while also multiplying to the constant term when the leading coefficient is 1.
So we can establish these equations
a+b = 23/5
a*b = -4/5


Let's square both sides of the first equation
a+b = 23/5
(a+b)^2 = (23/5)^2
a^2+2ab+b^2 = 529/25
a^2+2*(-4/5)+b^2 = 529/25 ......... plug in ab = -4/5
a^2-8/5+b^2 = 529/25
a^2+b^2 = 529/25+8/5
a^2+b^2 = 529/25+40/25
a^2+b^2 = 569/25
The motivation for this paragraph of algebra might not be obvious until reaching the next section below.


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Then,
{{{c = 1/(a^2) + 1/(b^2)}}}


{{{c*(ab)^2 = (1/(a^2) + 1/(b^2))*(ab)^2}}} Multiplying both sides by the LCD to clear out the fractions


{{{c*(ab)^2 = b^2 + a^2}}}


{{{c*(-4/5)^2 = 569/25}}}


{{{c*(16/25) = 569/25}}}


{{{c = (569/25)*(25/16)}}}


{{{c = 569/16}}}


{{{1/(a^2) + 1/(b^2) = 569/16}}}


I used GeoGebra to verify the answer is correct.
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