You need to find two numbers that multiply to give 72 and add up to give 18

easiest way: write the multiples of 72:

1, 72

2, 36

3, 24

4, 18

6, 12: these add up to 18

 (x + 6)(x + 12)

Nothing common cancels at the beginning. To factor this, we need to find two numbers that multiply to 9 * 4 = 36 and sum to 12. 6 and 6 work.

So 9x² + 12x + 4 = 9x² + 6x + 6x + 4

Let's look at the first two terms and last two terms separately to begin with. 9x² + 6x can be simplified to 3x(3x + 2) and 6x + 4 can be simplified into 2(3x + 2). Putting these together gets us 

9x² + 12x + 4

= 9x² + 6x + 6x + 4

= 3x(3x + 2) + 2(3x + 2) 

= (3x + 2)(3x + 2)

This is as far as we can factor. 

The numerator on the left can be factored so the expression becomes , which can be simplified to

Then you can solve for  by adding 3 to both sides of the equation, so 

First, factor.

Set each factor equal to 0

Therefore,

Let's try to factor x² – y² – z² + 2yz.

Notice that the last three terms are very close to y² + z² – 2yz, which, if we rearranged them, would become y² – 2yz+ z². We could factor y² – 2yz+ z² as (y – z)², using the general rule that p² – 2pq + q² = (p – q)² .

So we want to rearrange the last three terms. Let's group them together first.

x² + (–y² – z² + 2yz)

If we were to factor out a –1 from the last three terms, we would have the following:

x² – (y² + z² – 2yz)

Now we can replace y² + z² – 2yz with (y – z)².

x² – (y – z)²

This expression is actually a differences of squares. In general, we can factor p² – q² as (p – q)(p + q). In this case, we can substitute x for p and (y – z) for q.

x² – (y – z)² = (x – (y – z))(x  + (y – z))

Now, let's distribute the negative one in the trinomial x – (y – z)

(x – (y – z))(x  + (y – z)) 

(x – y + z)(x + y – z)

The problem said that factoring x² – y² – z² + 2yz would result in two polynomials in the form (ax + by + cz)(dx + ey + fz), where a, b, c, d, e, and f were all integers, and a > 0.

(x – y + z)(x + y – z) fits this form. This means that a = 1, b = –1, c = 1, d = 1, e = 1, and f = –1. The sum of all of these is 2.

The answer is 2. 

is a difference of squares.

The difference of squares formula is .

Therefore, = .

We can first factor out :

This factors further because there is a difference of squares:

You can solve this problem by substituting in  for  and solving for :

That means that  Fahrenheit is the same as  Celsius.

Group all the terms with the  variable.

Pull out an  term from parentheses.

There are no more common factors.

The correct answer is:  

This word problem can be broken down into a basic algebraic equation:

A half-loaded semi weighs 37,000 lbs. Subtracting the weight of the truck, we can determine that a half load weighs 17,000 lbs:

From that, we can determine that a full load = 34,000 lbs: 

Knowing the weight of a full load, we can calculate the weight of a 3/4 load:

Add the weight of the truck to get the total weight: