SOLUTION: Please l need your help for this assignment.if a and b are the root of the quadratic equation 3x³-4x-7=0, find a²/B +B²/a

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Question 1196712: Please l need your help for this assignment.if a and b are the root of the quadratic equation 3x³-4x-7=0, find a²/B +B²/a
Found 4 solutions by greenestamps, ikleyn, MathTherapy, math_tutor2020:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Don't mix upper case and lower case letters when naming your variables. "b" and "B" in the same problem can represent completely different quantities.

We are to evaluate the expression

a%5E2%2Fb%2Bb%5E2%2Fa

which, combining the fractions with a common denominator, is equivalent to

%28a%5E3%2Bb%5E3%29%2Fab

We can work this problem using Vieta's Theorem which says the sum of the roots is a%2Bb=4%2F3 and the product of the roots is ab=-7%2F3

%28a%5E3%2Bb%5E3%29%2Fab

Use the factorization pattern for the sum of cubes:

%28%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29%29%2Fab

Since we know the values of a+b and ab, rewrite the other term as

a%5E2-ab%2Bb%5E2=%28a%5E2%2B2ab%2Bb%5E2%29-3ab=%28a%2Bb%29%5E2-3ab

Then the expression we are to evaluate is all in terms of (a+b) and (ab):

%28%28a%2Bb%29%28%28a%2Bb%29%5E2-3ab%29%29%2Fab

Substitute a%2Bb=4%2F3 and ab=-7%2F3 and evaluate.

%28%284%2F3%29%28%284%2F3%29%5E2-3%28-7%2F3%29%29%29%2F%28-7%2F3%29

%28-3%2F7%29%28%28%284%2F3%29%28%284%2F3%29%5E2-3%28-7%2F3%29%29%29%29

%28-4%2F7%29%2816%2F9%2B7%29

-64%2F63-4

-316%2F63

ANSWER: a%5E2%2Fb%2Bb%5E2%2Fa=-316%2F63

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
Please l need your help for this assignment.
If a and b are the root of the quadratic equation 3x³-4x-7=0, find a²/b +b²/a.
~~~~~~~~~~~~~~~~ 

From the given part, we have, due to Vieta's formulas

    a + b = 4%2F3;  a*b = -7%2F3.     (1)


From the other side,

    a%5E2%2Fb + b%5E2%2Fa = %28a%5E3+%2B+b%5E3%29%2Fab.    (2)


The numerator is 

    a%5E3%2Bb%5E3 = %28a%2Bb%29%2A%28a%5E2+-+ab+%2B+b%5E2%29 = %28a%2Bb%29%2A%28%28a%5E2+%2B+2ab+%2B+b%5E2%29+-+3ab%29%29 = 

            = %28a%2Bb%29%2A%28%28a%2Bb%29%5E2-3ab%29 = now substitute from (1) = %284%2F3%29%2A%28%284%2F3%29%5E2+-+3%2A%28-7%2F3%29%29 = 

            = %284%2F3%29%2A%28%284%2F3%29%5E2+%2B+7%29%29 = %284%2F3%29%2A%2816%2F9%2B7%29 = %284%2F3%29%2A%28%2816%2B9%2A7%29%2F9%29 = %284%2F3%29%2A%2879%2F9%29 = 316%2F27.


Now we can continue and complete (2) in this way

    a%5E2%2Fb + b%5E2%2Fa = %28a%5E3+%2B+b%5E3%29%2Fab = %28%28316%2F27%29%29%2F%28%28-7%2F3%29%29 = - %28316%2A3%29%2F%2827%2A7%29 = - 316%2F63 = -51%2F63.    ANSWER

Solved.



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Please l need your help for this assignment.if a and b are the root of the quadratic equation 3x³-4x-7=0, find a²/B +B²/a
              
x(3x - 7) + 1(3x - 7) = 0
 x + 1 = 0     and      3x - 7 = 0, and so, roots, or 

                                                   

Switching the variables  yields the same result.

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

I think you meant to say 3x^2-4x-7 = 0 instead of 3x^3-4x-7 = 0

Compare 3x^2-4x-7 = 0 to ax^2+bx+c=0
to find that
a = 3, b = -4, c = -7
These a,b values are NOT the roots mentioned
I wish your teacher used p,q for the roots (or anything else except for a,b)

Anyways this is what the steps look like for the quadratic formula.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-4%29%2B-sqrt%28%28-4%29%5E2-4%283%29%28-7%29%29%29%2F%282%283%29%29

x+=+%284%2B-sqrt%28100%29%29%2F%286%29

x+=+%284%2B-++10%29%2F%286%29

x+=+%284%2B10%29%2F%286%29 or x+=+%284-10%29%2F%286%29

x+=+14%2F6 or x+=+-6%2F6

x+=+7%2F3 or x+=+-1
The two roots are p = 7/3 and q = -1
The order of the roots doesn't matter.

We're asked to compute %28p%5E2%29%2F%28q%29+%2B+%28q%5E2%29%2Fp where p,q are the roots of 3x^2-4x-7 = 0

%28p%5E2%29%2F%28q%29+%2B+%28q%5E2%29%2Fp

%28%287%2F3%29%5E2%29%2F%28-1%29+%2B+%28%28-1%29%5E2%29%2F%287%2F3%29

%28%2849%2F9%29%29%2F%28-1%29+%2B+%281%29%2F%287%2F3%29

-49%2F9+%2B+3%2F7

%28-49%2A7%29%2F%289%2A7%29+%2B+%289%2A3%29%2F%289%2A7%29

%28-343%29%2F%2863%29+%2B+%2827%29%2F%2863%29

%28-343%2B27%29%2F%2863%29

-316%2F63

Therefore,
%28p%5E2%29%2F%28q%29+%2B+%28q%5E2%29%2Fp+=+-316%2F63
when p = 7/3 and q = -1, which are the roots of 3x^2-4x-7 = 0