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.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> 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.      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!


2x%5E2+-+8x+%2B+7+=+-3x%5E2+%2B+15x+%2B+11

5x%5E2-23x-4=0

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

a=%2823%2Bsqrt%2823%5E2-4%285%29%28-4%29%29%29%2F10=%2823%2Bsqrt%28609%29%29%2F10
b=%2823-sqrt%2823%5E2-4%285%29%28-4%29%29%29%2F10=%2823-sqrt%28609%29%29%2F10

1%2Fa%5E2=100%2F%2823%5E2%2B609%2B46sqrt%28609%29%29=100%2F%281138%2B46sqrt%28609%29%29
1%2Fb%5E2=100%2F%2823%5E2%2B609-46sqrt%28609%29%29=100%2F%281138-46sqrt%28609%29%29

1%2Fa%5E2%2B1%2Fb%5E2=100%2F%281138%2B46sqrt%28609%29%29%2B100%2F%281138-46sqrt%28609%29%29


ANSWER: 569/16

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

Given the equation 5x%5E2-23x-4=0, Vieta's Theorem tells us

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

Rewrite the expression 1%2Fa%5E2%2B1%2Fb%5E2 in terms of (a+b) and (ab).



So



ANSWER (again, of course): 569/16

Answer by ikleyn(52781) About Me  (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 = 23%2F5,    (2)

    ab = -4%2F5.       (3)



Next,  

    1%2Fa%5E2 + 1%2Fb%5E2 = %28a%5E2+%2B+b%5E2%29%2F%28a%5E2%2Ab%5E2%29.    (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 23%2F5  and replace ab by  -4%2F5  based on (2),(3) and continue

              = %2823%2F5%29%5E2 - 2%2A%28-4%2F5%29 = 529%2F25 + 8%2F5 = %28529%2B5%2A8%29%2F25 = 569%2F25.


Therefore

    1%2Fa%5E2 + 1%2Fb%5E2 = %28a%5E2+%2B+b%5E2%29%2F%28a%5E2%2Ab%5E2%29 = %28%28569%2F25%29%29%2F%28%2816%2F25%29%29 = 569%2F16 = 359%2F16 = 35.5625.    ANSWER

Solved.



Answer by math_tutor2020(3817) About Me  (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,
c+=+1%2F%28a%5E2%29+%2B+1%2F%28b%5E2%29

c%2A%28ab%29%5E2+=+%281%2F%28a%5E2%29+%2B+1%2F%28b%5E2%29%29%2A%28ab%29%5E2 Multiplying both sides by the LCD to clear out the fractions

c%2A%28ab%29%5E2+=+b%5E2+%2B+a%5E2

c%2A%28-4%2F5%29%5E2+=+569%2F25

c%2A%2816%2F25%29+=+569%2F25

c+=+%28569%2F25%29%2A%2825%2F16%29

c+=+569%2F16

1%2F%28a%5E2%29+%2B+1%2F%28b%5E2%29+=+569%2F16

I used GeoGebra to verify the answer is correct.