SOLUTION: Let a and b be the solutions to 5x^2 - 11x + 4 = -x^2 + 5x - 3. Find a^2/b + b^2/a

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Question 1209218: Let a and b be the solutions to 5x^2 - 11x + 4 = -x^2 + 5x - 3. Find
a^2/b + b^2/a
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 520/63

Explanation

5x^2 - 11x + 4 = -x^2 + 5x - 3
rearranges to
6x^2 - 16x + 7 = 0
after getting everything to one side.

Then divide each term by the leading coefficient to arrive at
x^2 - (8/3)x + 7/6 = 0
This is so we have a leading coefficient of 1.

Due to Vieta's Theorems (specifically the quadratic versions), we know these two facts:
  • The roots add to the negative of the x coefficient.
  • The roots multiply to the constant term.
These rules apply only when the leading coefficient is 1.

Based on those two rules we can say
a+b = 8/3
a*b = 7/6
Let's call these equations (1) and (2)

Another useful equation we'll need is
a^2+b^2 = (a+b)^2 - 2ab
which is derived from
(a+b)^2 = a^2+2ab+b^2

Let's label the equation a^2+b^2 = (a+b)^2 - 2ab as equation (3).

--------------------------------------------------------------------------

%28a%5E2%29%2Fb+%2B+%28b%5E2%29%2Fa

= %28a%5E3%29%2F%28ab%29+%2B+%28b%5E3%29%2F%28ab%29

= %28a%5E3%2Bb%5E3%29%2F%28ab%29

= %28%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29%29%2F%28ab%29 Apply the sum of cubes factoring rule

= %28%28a%2Bb%29%28a%5E2%2Bb%5E2-ab%29%29%2F%28ab%29

= %28%28a%2Bb%29%28%28a%2Bb%29%5E2-2ab-ab%29%29%2F%28ab%29 Use equation (3)

= %28%28a%2Bb%29%28%28a%2Bb%29%5E2-3ab%29%29%2F%28ab%29

= %28%288%2F3%29%28%288%2F3%29%5E2-3%287%2F6%29%29%29%2F%287%2F6%29 Use equations (1) and (2).

= %288%2F3%29%2864%2F9-7%2F2%29%2A%286%2F7%29

= %288%2F3%29%28128%2F18+-+63%2F18%29%2A%286%2F7%29

= %288%2F3%29%2865%2F18%29%2A%286%2F7%29

= %288%2A65%2A6%29%2F%283%2A18%2A7%29

= 3120%2F378

= %286%2A520%29%2F%286%2A63%29

= 520%2F63

Therefore we determine that %28a%5E2%29%2Fb+%2B+%28b%5E2%29%2Fa+=+520%2F63 where a,b are the roots of 5x^2 - 11x + 4 = -x^2 + 5x - 3

I used GeoGebra to verify the solution is correct.

520/63 = 8.253968 approximately

More practice is found here

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Let a and b be the solutions to 5x^2 - 11x + 4 = -x^2 + 5x - 3.
Find a^2/b + b^2/a
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

The given equation, after reducing, is equivalent to this quadratic equation

    6x^2 - 16x + 7 = 0.    (1)


According to Vieta's theorem, if "a" and "b" are the roots of this quadratic equation, then

    a + b = 16%2F6 = 8%2F3,     (2)

      ab  = 7%2F6.           (3)


Now,  a%5E2%2Fb + b%5E2%2Fa = %28a%5E3%2Bb%5E3%29%2Fab.


The numerator is

    a^3 + b^3 = (a+b)*(a^2 - ab + b^2) = (a+b)*((a^2+2ab+b^2)-3ab) = (a+b)*((a+b)^2-3ab) = (a+b)^3 - 3(a+b)*(ab).


Now replace here  (a+b)  by   8/3  based on  (2)  and replace ab by  7/6  based on (3).

You will get  


    a^3 + b^3 = %288%2F3%29%5E3 - 3%2A%288%2F3%29%2A%287%2F6%29 = 512%2F27 - %288%2F3%29%2A%287%2F2%29 = 512%2F27 - 28%2F3 = %28512-9%2A28%29%2F27 = 260%2F27.


Therefore,  a%5E2%2Fb+%2B+b%5E2%2Fa = %28%28260%2F27%29%29%2F%28%287%2F6%29%29 = %28260%2A6%29%2F%2827%2A7%29 = 1560%2F189 = 520%2F63.


ANSWER.  a%5E2%2Fb+%2B+b%5E2%2Fa = 520%2F63.

Solved.