SOLUTION: Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = -3x^2 + 15x + 11. Compute 1/a^2 + 1/b^2.

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Question 1209228: Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = -3x^2 + 15x + 11. Compute 1/a^2 + 1/b^2.
Found 3 solutions by greenestamps, ikleyn, math_tutor2020:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!






First a slow algebraic method for finding the answer, using the quadratic formula to find the two roots....










ANSWER: 569/16

And now a MUCH easier solution, using Vieta's Theorem....

Given the equation , Vieta's Theorem tells us

(a+b) = 23/5
(ab) = -4/5

Rewrite the expression in terms of (a+b) and (ab).



So



ANSWER (again, of course): 569/16
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = -3x^2 + 15x + 11.
Compute 1/a^2 + 1/b^2.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

This given equation is reduced to the standard form quadratic equation

    5x^2 - 23x - 4 = 0.    (1)


Therefore, according to Vieta's theorem,  

    a + b = ,    (2)

    ab = .       (3)



Next,  

     +  = .    (4)


The numerator in (4) is

    a^2 + b^2 = (a^2 + 2ab + b^2) - 2ab = (a+b)^2 - 2ab = 

                replace here a+b by   and replace ab by    based on (2),(3) and continue

              =  -  =  +  =  = .


Therefore

     +  =  =  =  = 35 = 35.5625.    ANSWER

Solved.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 569/16
569/16 = 35.5625 exactly without any rounding done to it.

--------------------------------------------------------------------------
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.

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

Then,


Multiplying both sides by the LCD to clear out the fractions













I used GeoGebra to verify the answer is correct.
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