SOLUTION: If the ratio of a to b is equal to the ratio of b to a, what can you say about the two numbers a and b? Are they the same or different? Justify your answer. 1.Solve the followi

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: If the ratio of a to b is equal to the ratio of b to a, what can you say about the two numbers a and b? Are they the same or different? Justify your answer. 1.Solve the followi      Log On


   

Question 76884: If the ratio of a to b is equal to the ratio of b to a, what can you say about the two numbers a and b? Are they the same or different? Justify your answer.
1.Solve the following equation by using the quadratic formula/completing the square
3x^2 + x + 1 = 0


2.Solve the following equation by the method of your choice
2/3x-1 = 3/x+1
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
If the ratio of a to b is equal 
to the ratio of b to a, what can you 
say about the two numbers a and b? 
Are they the same or different? 
Justify your answer. 

a%2Fb+=+b%2Fa

Cross-multiply and get

a%5E2+=+b%5E2

Subtract b%5E2 from both sides: 

a%5E2-b%5E2=0

Factor %28a-b%29%28a%2Bb%29=0

a-b=0 a%2Bb=0

  a=b a=-b

So a and b are either the same or else
one is the other with the opposite sign.

The best way to say it is |a| = |b|,
that is, their absolute values
are equal.

1.Solve the following equation by using 
the quadratic formula/completing the square 

           3x² + x + 1 = 0

Using the quadratic formula:

                  ______ 
            -b ± Öb²-4ac
        x = —————————————
                2a 

where a = 3; b = 1; c = 1

                     _____________ 
             -(1) ± Ö(1)²-4(3)(1)
        x = ————————————————————————
                     2(3) 
                   ____ 
             -1 ± Ö1-12
        x = ————————————
                 6

                   ___ 
             -1 ± Ö-11
        x = ———————————
                6

                    __ 
             -1 ± iÖ11
        x = ————————————
                 6 

                      __
             -1     iÖ11
        x = ———— ± —————
              6      6
                      __ 
               1     Ö11
        x = - ——— ± ———— i
               6      6

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

Completing the square

           3x%5E2+%2B+x+%2B+1+=+0

Add -1 to both sides:

               3x%5E2+%2B+x+=+-1+

Multiply every term by 1%2F3

           %281%2F3%293x%5E2+%2B+%281%2F3%29x+=+%281%2F3%29%28-1%29

           x%5E2+%2B+%281%2F3%29x+=+-1%2F3

Multiply the coeff. of x, which is 1%2F3 by 1%2F2,
getting 1%2F6.  Then square that, getting 1%2F36,
then add 1%2F36 to both sides:

           x%5E2+%2B+%281%2F3%29x+%2B+1%2F36+=+-1%2F3%2B1%2F36

Factor the left side.  Get LCD on the right

           %28x+%2B+1%2F6%29%28x+%2B+1%2F6%29+=+-12%2F36%2B1%2F36

Write left side as the square of a binomial, and
combine the terms on the right:

           %28x%2B1%2F6%29%5E2+=+-11%2F36

Take square roots of both sides:

           x%2B1%2F6 = ±sqrt%28-11%2F36%29

           x%2B1%2F6 = ±sqrt%28-11%29%2Fsqrt%2836%29

           x%2B1%2F6 = ±i%2Asqrt%2811%29%2F6

           x+=+-1%2F6±+%28%28sqrt%2811%29%29%2F%286%29%29i 
        

2.Solve the following equation by the method 
of your choice:

2%2F%283x-1%29+=+3%2F%28x%2B1%29

Cross-multiply:

2%28x%2B1%29+=+3%283x-1%29

  2x%2B2+=+9x-3

  -7x+=+-5

  x=%28-5%29%2F%28-7%29

  x+=+5%2F7

Edwin