I need help

Question

a) Solve 3x² + 7x – 6= 0

 

(I can do this)

 

b) Hence state the positive value of y for which 3/y² + 7/y – 6= 0. 

 

(I can't understand this part. the solution shows that 1/y= 2/3 but why? I don't get it at all. Please help, thank you!)

Answer 1

3/y² + 7/y -6 = 0       3 + 7y - 6y² = 0    6y² - 7y -3 = 0

 

y = - (-7) √ (-7)² - 4(6) (-3) / 2 (6)

 

y₁ = 7 √ 121 / 12

y_{1 = (7 + 11) / 12}

y_{1 = 18 / 12}

y_{1 = 3 /2      is equal to    1 /y = 2 / 3}                 

Answer 2

THX for including "I can do this;" that's good so far!  So, what you need is "perspective."

 

Do these two example equations have anything in common:

      x = 5

      1/y = 5

Yes, since equal values equal each other, x = 1/y.

 

What about:

     x² + x + 1 = 0

    (1/y)² + (1/y) + 1 = 0

Well,

    x² + x + 1 = (1/y)² + (1/y) + 1

 

Now, we know that:

     (x² - 1) = (x+1)(x-1)             [either factor or F-O-I-L)

 so, what are the factors of:

     ((1/y)² - 1)    ??

 

Then, how do these equations relate to each other?

    3x² + 7x – 6= 0
   (3/y)² + 7(1/y) – 6= 0

  Specifically, if we know the solution (that is, the values of x for which equals is true) for the first equation, do we automatically [I mean really fast] know the solution to the second equation?  Well, O.K., you may want to check it a couple times until you are comfortable saying, "automatically."

 

Now, please feel free to do all the intermediate math until you are comfortable, but, hopefully, there will soon come a time when you will call this "TMI," or else TMW (Too Much Work).

 

Extra:  Why would the question say, "positive value?" 

  and,  Is there any difference between the allowable x-values and the allowable y-values?

  

Answer 3

You can do it this way too.

 

 

Multiply both sides of the equation by y² to eliminate any denominators.

 

3 + 7y - 6y² = 0

 

 

Next, we can use FOIL to factor out the left side of the equation.

 

(3 - 2y)(1 + 3y) = 0

 

 

Set the first factor equal to zero and solve for y, since that factor will lead to a positive solution.

 

3 - 2y = 0      

 

   -2y = -3

 

      y = 3/2

 

 

Now raise both sides of the equation by the power of -1 to get 1/y.

 

y^-1 = (3/2)^-1

 

1 / y = 1 / (3 / 2)

 

1/y = 2/3

Answer 4

(b)

 

Use the substitution x = 1/y.  Then

 

3/y² + 7/y - 6 = 0 becomes 3x² + 7x – 6 = 0, which you already know how to solve:

 

x = {-7 ± √[7² - 4*3*(-6)]} / (2*3) = [-7 ± √(49 + 72)] / 6 = (-7 ± 11) / 6 = 2/3 or -3

 

Then x = 1/y = 2/3 or -3

 

The positive root is 1/y = 2/3