Question

What is the simplified form of the following?

Answer 1

\[\frac{ 4 }{ 3y-2 }-\frac{ 5 }{ 9y^2 }\]

Answer 2

@Hero @ash2326

Answer 3

These can be tricky. As you know, in order to simplify this, we need to combine the fractions. However, as you already know, the denominators of both fractions must be the same before doing so.

Answer 4

In order to make the denominators the same, we have to multiply the top and bottom of each fraction by the appropriate factors so that each fraction has the same denominator.

We have
\[\frac{ 4 }{ 3y-2 }-\frac{ 5 }{ 9y^2 }\]

But \(9y^2\) is the factor missing from the first fraction so we have to multiply the top and bottom of the first fraction by \(9y^2\)

\[\frac{9y^2}{9y^2} \dot\ \frac{ 4 }{ 3y-2 }-\frac{ 5 }{ 9y^2 }\]

Answer 5

yeah, I got 9y^2-4 as the common denominator... Is that correct?

Answer 6

You may have misinterpreted what I was trying to show you.

Answer 7

You multiply the top and bottom of the fraction by \(9y^2\) and after multiplication, you end up with

\[ \frac{ 36y^2 }{ (9y^2)(3y-2) }-\frac{ 5 }{ 9y^2 }\]

Answer 8

Now we have to multiply top and bottom of the second fraction by its missing factor \((3y - 2)\)

Answer 9

\[\frac{ 36y^2 }{ (9y^2)(3y-2) }-\frac{ 5 }{ 9y^2 } \dot\ \frac{(3y - 2)}{(3y - 2)}\]

Answer 10

Afterwards we end up with:

\[\frac{ 36y^2 }{ (9y^2)(3y-2) }-\frac{ 15y-10 }{ 9y^2(3y - 2)}\]

Answer 11

Now we can combine the fraction since they both have the same denominator:

\[\frac{ 36y^2 -(15y - 10)}{ (9y^2)(3y-2) }\]

Answer 12

Well, but we're not done

Answer 13

We can expand the numerator to get:

\[\frac{ 36y^2 -15y + 10}{ (9y^2)(3y-2) }\]

Answer 14

The \(36y^2 - 15y + 10\) is likely factorable

Answer 15

I don't think it is....

Answer 16

Yes, you're right since the discriminant yields (-15)^2 - 4(36)(10) = -1215 which is not perfect square and even if it was, it is negative.