SOLUTION: FInd the quadratic equation each of whose roots is the sum of a root and its reciprocal of the quadratic equation 2x^2 + 3x + 4 = 0.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: FInd the quadratic equation each of whose roots is the sum of a root and its reciprocal of the quadratic equation 2x^2 + 3x + 4 = 0.      Log On


   

Question 1183884: FInd the quadratic equation each of whose roots is the sum of a root and its reciprocal of the quadratic equation 2x^2 + 3x + 4 = 0.
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
FInd the quadratic equation each of whose roots is the sum of a root
and its reciprocal of the quadratic equation 2x^2 + 3x + 4 = 0.
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            This problem is to apply the  Vieta's theorem several times. 


Let  " a "  and  " b "  be the roots of the given equation.


Then according to Vieta's theorem, the sum  (a+b)  is equal to  -3%2F2 :  a + b = -3%2F2.

According to the same theorem, the product of the routs  ab  is equal to  4%2F2 = 2.



                OK.  



HENCE, the projected equation has the value  -3%2F2  as one of its roots.

The other root of the projected equation is  the sum of reciprocals  1%2Fa%2B1%2Fb = %28a%2Bb%29%2F%28ab%29.


In this expression, we can replace  a+b  by  -3%2F2  and can replace  ab  by  2, since we just know it.


Making this replacement, you get  1%2Fa%2B1%2Fb = %28%28-3%2F2%29%29%2F2 = -3%2F4.


So, the roots of the projected equation are  -3%2F2  and  -3%2F4.


Hence, the sought equation is


    %28x-%28-3%2F2%29%29%2A%28x-%28-3%2F4%29%29 = 0,   or, EQUIVALENTLY,  %28x%2B3%2F2%29%2A%28x%2B3%2F4%29 = 0.


If you want to have the sought equation with integer coefficients, you can multiply the last equation by 8,

leaving it equivalent.


In this way, you get the ANSWER:


    the sought equation is  (2x+3)*(4x+3) = 0  with the roots  of  -3%2F2  and  -3%2F4.

Solved.

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There is another way to solve the problem.

It is to find the roots of the given equation explicitly and then calculate their sum and the sum of reciprocals.

It is possible,  but it will lead you through the forest of calculations with radicals and fractions of complex numbers with radicals.

So,  this way is possible,  but it is not elegant.

The way which I showed you in my post,  is  TRULY  ELEGANT,  and   IT   IS   the  ONLY  expected way
as the problem should be solved and presented.


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May I ask you ? 

    This problem is advanced: it is destined for advanced students.

    Advanced students are THOSE who love Math and love solving tricky problems on their own.

    If so, then what is the reason for you, for an advanced student, post this problem
    to the forum instead of solving it on YOUR OWN ?