ilorty

Completing the square: a very useful technique. 
 

Hmm..that sounds like a good book title

Anyways, we know that circles are in the form (x−a)^2 + (y−b)^2 = r^2, so how do we get there? By completing the square! (Wow, that rhymes...)

We need to deal with both the x and y.

X: 

We see that x^2 - 18x is a fraction of a square. To find the last bit, we divide -18 by 2: this gives us -9. So, we need to take away 81 from 120. This leaves us with: (x−9)^2 + (y−b)^2 + 39 = 0.

Y: 

Again, our incomplete square is y^2 - 14y. Divide by 2: -7. Subtract 49: leaves us with -10. Which makes our equation: (x−9)^2 + (y−7)^2 - 10 = 0.

(x−9)^2 + (y−7)^2 - 10 = 0

(x−9)^2 + (y−7)^2 = 10

Therefore, the radius is sqrt(10)

:)

This is just a fancy way of saying: Find cos(arctan(3/4)).                                                                                                                           
                                                                                                                 1

Like most things in math, there is a formula for it: cos(arctan(x)) = -----------------                                                                                                                                                                 

                                                                                                         sqrt(1 + x^2)   

So...what do we do? Well, we plug in 3/4.

            1                            1                 4  

---------------------- = -------------- = ------ 

 sqrt(1 + (3/4)^2)              5/4               5

After simplification, we realize two things:

1. The answer is 4/5

2. Typing fractions by hand is hard

:)

Don't worry. When the problem has eyes, I always get nervous

(get it? Eyes, i's? Haha, dad joke made from a "never-will-be" dad)

It's as simple as 66 + 57. (123)

All you have to do is plug the values in.

4 - 2i - 2(3 + 2i)

4 - 2i - 6 - 4i

-2 - 6i

There! 3 steps, and you are done.

:)

There are three ways to do this.

Method 1: Use a graph

The closer two numbers are when added together, the greater their product. Here is what I mean. 
 

Let's "shrink" this question a bit by changing rectangle to two lines, and change 60 into 10. Okay. So how would we get the largest value? We can do some guess and checking...assuming that you don't know the rule yet.

Comparing 1, 9 and 5, 5, we see that 5 * 5 is larger. Get what I mean now?

The same thing applies to 4 sides, and...that's about it. (BUT if we were talking about numbers, then it would go past the skies)

The closest the number can get to each other is 15 by 15. Simple enough, right? 

Therfore, the maximum area is 15 * 15 = 225

:)

Just telling you, guest did not go into your computer and steal all those files.

What he/she did was long press (if on mobile) the image, or left click (if on PC or MAC) and select "copy image". This will give the link to the image, which gives the picture.

I think the reason why guest's pictures were not blocked (or sent for moderation) was because they were the web2.0calc links, which the system probably recognizes as 'innocent'. Maybe.

Have a good day Gwen!

:)

This can be done algebraicly, and without what seems to look like a not understandable copy and pasted answer.

Using some factoring:

6xyz + 30xy + 21xz + 2yz + 105x + 10y + 7z = 812

3x(2yz + 10y + 7z + 35) + (2yz + 10y + 7z) = 812      <------ factor out 3x, and "join like terms"

We see that the difference between the first and the second is "35", so we add 35 to both sides:

3x(2yz + 10y + 7z + 35) + (2yz + 10y + 7z + 35) = 847 

We can factor out a 2y and a 7 from the left:

3x(2y + 7)(z + 5) + 1(2y + 7)(z +5) = 847

Now, we can factor out (2y + 7)(z + 5).

(3x + 1)(2y + 7)(z + 5) = 847

The prime factorization of 847 is 7 * 11 * 11, which can be applied to our factored form (they are both factors....get it?):

So: 

3x + 1 = 7: x = 2

2y + 7 = 11: y = 2

z + 5 = 11: z = 6

Therefore, x + y + z = 10

:)

Since angles that are across from each other (formed by two 180 degree lines: anyone know what the term is? I forgot...), we can say that:

6x - 3 = 2x + 13

6x = 2x + 16

4x = 16

x = 4

Therefore, x = 4.

:)

Im sorry. I was really skeptical about the other answer....but I didn't have a better way...so....