Question 1209123
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Answer: <font color=red>-33/16</font>



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Explanation


I'll use p,q in place of a,b
This is because a,b,c are the standard coefficients of the quadratic template {{{ax^2+bx+c = 0}}}.
In the case of x^2+7x-4 = 0 we have a = 1, b = 7, c = -4.


Instead of computing {{{(a + 3)/(b + 3) + (b + 3)/(a + 3)}}} the expression I'll evaluate is {{{(p + 3)/(q + 3) + (q + 3)/(p + 3)}}}


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I'll take a slight detour for a moment.


From the quadratic version of Vieta's Formulas, we know that:
p+q = -b/a
p*q = c/a
When plugging a = 1, b = 7, and c = -4, we get
p+q = -b/a = -7/1 = -7
p*q = c/a = -4/1 = -4


In short,
p+q = -7
p*q = -4
Let's call these equation (1) and equation (2) to be used later.


Then note the following
{{{(p+q)^2 = p^2+2pq+q^2}}}


{{{p^2+q^2 = (p+q)^2-2pq}}}


{{{p^2+q^2 = (-7)^2-2(-4)}}} Applying equations (1) and (2)


{{{p^2+q^2 = 57}}} Let's call this equation (3)



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Let's return to {{{(p + 3)/(q + 3) + (q + 3)/(p + 3)}}}
We'll combine the fractions. 
Recall we need the LCD to do so.


{{{(p + 3)/(q + 3) + (q + 3)/(p + 3)}}}


= {{{((p + 3)(p+3))/((p+3)(q + 3)) + ((q+3)(q + 3))/((p + 3)(q+3))}}}


= {{{(p^2+6p+9)/(pq+3p+3q+9) + (q^2+6q+9)/(pq+3p+3q+9)}}}


= {{{(p^2+6p+9+q^2+6q+9)/(pq+3p+3q+9)}}}


= {{{(p^2+q^2+6(p+q)+18)/(pq+3(p+q)+9)}}}


= {{{(highlight(p^2+q^2)+6(highlight(p+q))+18)/(highlight(pq)+3(highlight(p+q))+9)}}}


= {{{(highlight(57)+6(highlight(-7))+18)/(highlight(-4)+3(highlight(-7))+9)}}} Apply equations (1) through (3)


= {{{(57+6(-7)+18)/(-4+3(-7)+9)}}}


= {{{33/(-16)}}}


= {{{-33/16}}}


Therefore,
{{{(p + 3)/(q + 3) + (q + 3)/(p + 3) = -33/16}}}
where p,q are the roots of {{{x^2+7x-4=0}}}


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To verify, you can use the quadratic formula to solve {{{x^2+7x-4=0}}}
You should get {{{p = (-7+sqrt(65))/2}}} and {{{q = (-7-sqrt(65))/2}}} as the two roots.


Then plug each value into {{{(p + 3)/(q + 3) + (q + 3)/(p + 3)}}} and simplify. 


I used GeoGebra to verify the answer. 
Here's the link to that calculation
<a href="https://www.geogebra.org/calculator/fwzwpynj">https://www.geogebra.org/calculator/fwzwpynj</a>
Let me know if you have any questions.
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