SOLUTION: Solve these two equations: (a) The quadratic equation {{{2ax^2-4ax+a+1=0}}} has two roots. If one root is five times the other, what is the value of a? (b) If {{{x+2}}} and {

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Solve these two equations: (a) The quadratic equation {{{2ax^2-4ax+a+1=0}}} has two roots. If one root is five times the other, what is the value of a? (b) If {{{x+2}}} and {      Log On


   

Question 1195204: Solve these two equations:
(a) The quadratic equation 2ax%5E2-4ax%2Ba%2B1=0 has two roots. If one root is five times the other, what is the value of a?
(b) If x%2B2 and x-3 are factors of the polynomial p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb, find a.
Found 3 solutions by ikleyn, lotusjayden, MathTherapy:
Answer by ikleyn(52812) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve these two equations:

(a)   The quadratic equation 2ax%5E2-4ax%2Ba%2B1=0 has two roots.
        If one root is five times the other, what is the value of a?

(b)   If x%2B2 and x-3 are factors of the polynomial p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb,
        find a.
~~~~~~~~~~~~~~~~~~

            Part (a)


Let z be one root of the equation; then 5z is another root.


Apply the Vieta's theorem.  it gives you these two equations for unknowns "a" and "z":


    (1)  the sum of the roots z and 5z is equal to -%28-4a%29%2F%282a%29 = 2

             z + 5z = 2,  or  6z = 2,  z = 2%2F6 = 1%2F3.


    (2)  the product of the roots is equal to %28a%2B1%29%2F%282a%29

             z*(5z) = %28a%2B1%29%2F%282a%29,  which implies  %281%2F3%29%2A%285%2F3%29 = %28a%2B1%29%2F%282a%29,

             5%2F9 = %28a%2B1%29%2F%282a%29,  5*(2a) = 9*(a+1),  10a = 9a + 9,  a = 9.


Thus we get the ANSWER :  a = 9.

Part  (a)  is solved.


///////////////


Two problems in your post are different.

At this forum,  I do not recommend to pack more than one problem to post, in order for do not create a  MESS.

Therefore,  post your second problem separately,  and I will show you how to solve it.


By the way, the rules of this forum  PROHIBIT  posting more than one problem at once.

The rules are at this web-page

https://www.algebra.com/tutors/students/ask.mpl?action=ask_question&topic=Equations&return_url=http://www.algebra.com/algebra/homework/equations/

from which you post your problems.


It is assumed that you read these rules before posting.

It is also assumed that you do understand what is written in that page and follow the rules.


Those who violate them,  work against their own interests.



Answer by lotusjayden(18) About Me  (Show Source):
You can put this solution on YOUR website!
I'll solve this part b by myself.
(b)   If x%2B2 and x-3 are factors of the polynomial p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb,
        find a.
~~~~~~~~~~~~~~~~~~ 
First, you should multiply the two known factors together: 

%28x%2B2%29%28x-3%29

which simplifies to:
x%5E2-x-6

after you get that, one method to solve this problem is to do trial and error:

After a while, you should get:

%28x%5E2-x-6%29%28x%2B6%29

The simplified version is:
x%5E3%2B5x%5E2-12x-36

Now, you can see that a=-12, and b=-36.

Thus we get the ANSWER :  a = -12.

   Solved.



Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
Solve these two equations:
(a) The quadratic equation 2ax%5E2-4ax%2Ba%2B1=0 has two roots. If one root is five times the other, what is the value of a?
(b) If x%2B2 and x-3 are factors of the polynomial p%28x%29+=+x%5E3%2B5x%5E2%2Bax%2Bb, find a.
matrix%281%2C3%2C+2ax%5E2+-+4ax+%2B+a+%2B+1%2C+%22=%22%2C+0%29 =====> matrix%281%2C4%2C+%282a%29x%5E2+%28-+4a%29x%2C+%22+%2B%28a+%2B+1%29%22%2C+%22=%22%2C+0%29

Let smaller root be R
Then larger root = 5R
Sum of the roots, or . Also, sum of roots = R + 5R = 6R.
                                  Therefore, 

Product of the roots, or matrix%281%2C5%2C+p%2C+%22=%22%2C+c%2Fa%2C+%22=%22%2C+%28a+%2B+1%29%2F%282a%29%29. Also, product of roots = R(5R) = 5R2 = matrix%281%2C5%2C+5%281%2F3%29%5E2%2C+%22=%22%2C+5%281%2F9%29%2C+%22=%22%2C+5%2F9%29 
                       Therefore, matrix%281%2C3%2C+%28a+%2B+1%29%2F%282a%29%2C+%22=%22%2C+5%2F9%29
                                     10a = 9(a + 1)----- Cross-multiplying
                                     10a = 9a + 9
                                10a - 9a = 9
                                      highlight_green%28matrix%281%2C3%2C+a%2C+%22=%22%2C+9%29%29
***BTW, this is Part (a).