SOLUTION: If a and b are the roots of the quadratic equation 3x^2 - 4x + 7 =0 , find: 1) a/3b + b/3a 2) 1/a^2 + 1/b^2 Thank you :)

Algebra.Com
Question 1010881: If a and b are the roots of the quadratic equation 3x^2 - 4x + 7 =0 , find:
1) a/3b + b/3a
2) 1/a^2 + 1/b^2

Thank you :)
Found 3 solutions by Boreal, MathLover1, ikleyn:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
3x^2 - 4x + 7 =0
x=(1/6){4 +/- sqrt (16-84); sqrt term =2i sqrt (17)
x=(1/6)(4+/- 2i sqrt (17)
x=(2/3)+/- (i/3)sqrt(17)
I will let a= 2/3+(i sqrt(17)/3)
b=2/3 - (i sqrt (17)/3)
a/3b + b/3a=
{2+isqrt (17)/3/2-isqrt(17)} + {2-isqrt (17)/3/2+isqrt(17)}
common denominator is (2-i sqrt (17)) (2+ i sqrt (17))=4-17i^2=4+17=21
the first term becomes (1/3)4+4i sqrt (17)-13); the second term is (1/3)(4-4i sqrt (17)-13)
They add to (1/3){8-26), because the middle terms cancel out, and that is (1/3)(-18)= -6
-6/21=-2/7 ANSWER
a^2= (1/3)*(2+i sqrt 17) all squared.
That is (1/9){4+4i sqrt(17)-17)=(1/9)(-13+4i sqrt(17)
1/a^2 = 9/-13 +4i (sqrt 17))
b^2= (1/3)(2-i sqrt (17) all squared
that is (1/9)(4-4 i sqrt (17)-13=(-13-4i sqrt (17)
1/b^2=9/(-13-4i sqrt (17)
common denominator is the conjugate, and that is 169+262=431
the numerator will be 9(-13+4i sqrt (17)) + 9 (-13-4i sqrt (17)=-234
-234/431



Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!
If and are the roots of the quadratic equation , find:
1)
2)
first find the roots and :
.....use quadratic formula








so,




1)





















....exact

or approximately



2)









....
exact or approximately



Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
If a and b are the roots of the quadratic equation 3x^2 - 4x + 7 =0 , find:
1) a/3b + b/3a
2) 1/a^2 + 1/b^2
------------------------------------------------------------------------- 
If a and b are the roots of the quadratic equation  =  , then

a + b =      (1))   and 

ab = .       (2)

It follows the general theorem:

    if a and b are the roots of the quadratic equation  =  then  =  and ab = .

From (1) and (2),  =  =  =  =  =  =  = .  (3)

    Notice that we got that the sum of two squares, , is negative. 
    It means that the roots a and b are complex numbers, not real roots. 
    But this fact does not reject our calculations.

Hawing this, we can easily calculate , which is #1 of your claim. It is

 =  =  =  :  =  :  = .

Next is #2 of the claim:

 =  =  =  :  =  :  =  = .

So, we do not need to calculate the roots of the given equation separately and individually to answer the claim. 
All the required information is just contained in the coefficients of the equation.

This is the major idea I want to bring to you.
This idea is the key to solving this problem and other problems similar to it.


RELATED QUESTIONS

If the two roots of the quadratic equation x^2-6x+1=0 are log a and log b, what is the... (answered by KMST)
The roots of the equation 4x^2-x+36=0 are (alpha)^2 and (beta)^2. Find: (a) an equation (answered by josmiceli)
im really stuck on this question the roots of the quadratic equation x^2 + 4x - a= 0... (answered by jsmallt9)
Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = -3x^2 + 15x + 11. Compute... (answered by greenestamps,ikleyn,math_tutor2020)
If a and b; are roots of the equation {{{ ax ^ 2 + bx + c = 0 }}} , find the values of: (answered by ikleyn)
if tan a and tan b are the roots of equation 4x^2-7x+1=0 then evaluate... (answered by ikleyn)
The roots of the quadratic equation z^2 + az + b = 0 are 2 - 3i and 2 + 3i. What is a+b?... (answered by josgarithmetic)
Find the value of k on x^2+3x+k-1=0 if the quadratic equation has: a) equal roots b)... (answered by Fombitz)
Find the value of k in x^2+3x+k-1=0 if the quadratic equation has: a) equal roots b)... (answered by Fombitz)