Question

Question #c8810

Answer 1

If the question is how to factor this:

`(3y - 6)(3y - 7)`

If not, you need to let us know what you are looking to solve for.

Explanation:

I hope that helps!

Answer 2

`3(3y-7)(y-2)`

or

`x=3((-3 +- 1)/(6))`

(it depends on method of solving)

Explanation:

NOTE: This will be a long, detailed explanation involving two methods to simplify this expression. Be prepared.

Because you have the phrase equal to "?", I assume that this is an expression, and will simplify it to the point where it can be solved (if needed)

`9y^2-39y+42`

Now, to solve this, we could either factor or use the quadratic formula. Let's try factoring first.

FACTORING:

Now, the first thing that we can see is that this will be hard to factor, so let's find the greatest common factor in the equation. In this case, it happens to be 3, so we can factor that out:

`9y^2-39y+42`
=> `(3*3)y^2-(3*13)y+(3*14)`
=> `3(3y^2-13y+14)`

Now, we factor. This is tricky, as there is a coefficient on the `y^2`, but because it is prime, we can immediately start setting up the factoring as so:

`3(3y ) (y )`

Remember that the `3` we factored out must be present; it hasn't been simplified out, so it is still there.

Now, to determine the signs inside of the parentheses, we must look at the second and third terms of the expression `3y^2-13y+14`

`-13y+14`

The `14` is a result of two numbers being multiplied to equal `14`; because it is a positive `14`, both factors, whatever they are, must be both either positive or negative.

To determine whether they are positive or negative, however, we look at the second term, specifically the coefficient `-13`. This is the result of addition or subtracting numbers in the FOIL process (the opposite of factoring). Because it is negative, there must be one or more negative numbers involved; now we know that both signs must, in fact, be negative.

At this point, we are at this stage in the factoring:

`3(3y- ) (y- )`

Now, we must find the two numbers that must go inside the parentheses, which both multiply to `14`, and subtract to be `-13`

The only 2 possible combinations, in this case, are `-1,-14` and `-2,-7`; we are limited by the possible factors of the last number, and we established that the numbers must be negative (accounted for by the `-` signs in the parentheses). We can ignore the `3` outside for now, as it only needs to be used in the end). We can guess and check, by first plugging in `-1,-14`. The positioning of the numbers matters, as one will be multiplied by `3`.
`3(3y-14 ) (y-1 )`
`= 3(3y^2 -3y -14y +14)`
`= 3(3y^2 -17y+14)`

Though this answer is close, it is not correct; the first and last terms match up, but the middle term does not. Even if we did switch the order of the numbers, we would get the incorrect answer. Now, let us try the other possibility: `-2,-7`:

`3(3y-7 ) (y-2 )`
`= 3(3y^2-6y-7y+14)`
`= 3(3y^2-13y+14)`

Now, as we can see, this is the correct answer, as all three terms are equivalent to those of the starting terms. Now, our expression looks like this when simplified:

`3(3y-7)(y-2)`

And that is the final answer, simplified through Factoring!

QUADRATIC FORMULA:

Now, this is the more formulaic approach, though it does require a calculator. For this method, we plug in corresponding numbers to the quadratic formula, which is as follows:

`x=(-b +- sqrt(b^2-4ac))/(2a)`

where `+-` is `+` or `-`

Now, we can plug in the equation `9y^2-39y+42`, as it is in the correct form to do so; the correct form for an equation to be in to plug into the quadratic equation is:

`ax^2+bx+c`

Where `a` and `b` are the coefficients, and `c` is a constant.

So, in this case, the corresponding coefficients would be such:

`ax^2+bx+c`
`9y^2-39y+42`

`a = 9`
`b = -39`
`c = 42`

Now, we can plug in these values, and simplify as such:

`x=(-b +- sqrt(b^2-4ac))/(2a)`

=> `x=(-(9) +- sqrt((-39)^2-4(9*42)))/(2(9)`

=> `x=(-9 +- sqrt(1521-4(378)))/(18)`

=> `x=(-9 +- sqrt(1521-1512))/(18)`

=> `x=(-9 +- sqrt(9))/(18)`

=> `x=(-9 +- 3)/(18)`

We can simplify this even further by factoring out 3, which results in:

=> `x=((3*-3) +- (3*1))/(3*6)`

=> `x=3((-3 +- 1)/(6))`

And that is the final answer, through the quadratic formula!