All Answers 358134 Answers

You are correct, NinjaDevo.

Combining like terms, we get

7x^2 - 2xy + 17y^2

Yep....unlike the "keycode" problem..... in which the number of different answers might actually be approaching the same limit as the number of actual keycodes......(LOL!!!)

Always good to have a couple of different approaches (even better when they give the same answers!).

Let the initial size be N₀.

After 8 years N = 2*N₀.

After 2*8 =16 years N = 2*2*N₀ = 4N₀.

After 4*8 = 32 years N = 2*4*N₀ = 8N₀.

After 8*8 = 64 years N = 2*8*N₀ = 16N₀.

So it seems like the pattern is:

After n*8 years N = 2*n*N₀.

When n*8 = 40, we can see that n must be 5 (since 5*8 = 40) and we are told N = 100,000 so:

100000 = 2*5*N₀.

N₀ = 100000/10 or N₀ = 10,000

10 years from now is 50 years from the start, so n*8 = 50, n = 50/8

N = 2*(50/8)*10000 = 125,000

Golden Leaf's last line - where he divided through by 2 -  would be the correct approach if we were trying to solve something set to 0.......As it is, his next-to-the last line is the correct answer.....

Any x will work......absolute value is always either positive (or 0) and that will always be greater than -2 !!!

In "math" language, the solution is all real numbers or just (-∞, ∞)

Fooled me....looks pretty professional  !!!

Alan has an interesting approach.......mine is a little different........but it should still work out the same...

Let's divide the original equation by 4....so we have

x^2 + x + y^2 - 4y - 127/4 =0          add  127/4 to both sides

x^2 + x + y^2 - 4y = 127/4                now, complete the sqaure on x and y

(x^2 + x + 1/4) + (y^2 - 4y + 4) =  127/4 + 1/4 + 4         simplifying, we have

(x + 1/2)^2  + (y - 2)^2  = 144/4

(x + 1/2)^2 + (y - 2 )^2 = 36

So. the center is (-1/2, 2) and the radius is 6

(x-1)(2x^3+2)

FOIL - First Outer Inner Last

(x * 2x^3) (x * 2) (-1 * 2x^3) (-1 * 2)

2x^4 + 2x - 2x^3 - 2

2x^4 - 2x^3 + 2x - 2

This one I actually drew up this one myself, that's why it doesn't look the best!

Thanks for the compliment though. :)

I'm gonna' hire you as my "graphic artist"......ND.....that's a really neat picture.....(even if you did "borrow" it from another source!!!)

Thanks, ND.......I should have been more explicit in explaining how I got my answer (like you did!!!)

I gave you some points and a thumbs-up, too!!!

It's fine, Ninja:) You saved me the work there

sqrt(5) * sqrt(3) is the same thing as sqrt(5*3). There is a property of square roots that $${\sqrt{{\mathtt{x}}}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{y}}}} = {\sqrt{{\mathtt{x}}{\mathtt{\,\times\,}}{\mathtt{y}}}}$$ 

So simplified, the answer would be  $${\sqrt{{\mathtt{15}}}}$$

Oh, sorry kitty.

I didn't realize that you said you would show them how to solve this without a calculator in your post, so I went ahead and posted mine.

Didn't mean to steal your thunder there 

Kitty is right, if you are allowed to use a calculator to solve this question, it's often the quickest way to solve it.

However, if you wondering how you solve this question, I'll show you how to solve it so hopefully you can solve problems like this in the future without help. :)

Start by doing

$$\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{12}}}{{\mathtt{19}}}}\right)$$

We had to start by multiplying these two numbers because they were in parentheses, and based on the order of operations, you have to do parentheses before addition.

Divide 3 and 12 by 3 and we get

$$\left({\frac{{\mathtt{1}}}{{\mathtt{1}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{4}}}{{\mathtt{19}}}}\right)$$

$$\left({\frac{{\mathtt{4}}}{{\mathtt{19}}}}\right)$$

Now we need to add 1/4 to 4/19

$${\frac{{\mathtt{4}}}{{\mathtt{19}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}}$$

The problem here is that we dont have a common denimator, so we can't add these numbers together yet.

To get a common deminator, multiply the top and bottom of the first fraction by 4, and the top and bottom of the second fraction by 19.

$$\left({\frac{{\mathtt{4}}}{{\mathtt{19}}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{4}}}{{\mathtt{4}}}}\right)\right){\mathtt{\,\small\textbf+\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{4}}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{19}}}{{\mathtt{19}}}}\right)\right)$$

And we get

$${\frac{{\mathtt{16}}}{{\mathtt{76}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{19}}}{{\mathtt{76}}}}$$

Add them together:

$${\frac{{\mathtt{35}}}{{\mathtt{76}}}}$$

This fraction can't be reduced, so this is our final answer.

(2x² + 3)²

= (2x² + 3) * (2x² + 3)

Use the Distributive Property to expand:

= (2x² * 2x²) + (3 * 2x²) + (2x² * 3) + (3 * 3)

= 4x⁴ + 6x² + 6x² + 9

= 4x⁴ + 12x² + 9

You can plug this into this site's calculator:

$$\left({\frac{{\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,\times\,}}{\mathtt{12}}}{{\mathtt{19}}}}\right){\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}} = {\frac{{\mathtt{35}}}{{\mathtt{76}}}} = {\mathtt{0.460\: \!526\: \!315\: \!789\: \!473\: \!7}}$$ 

If, however, you meant how to solve this equation without a calculator, please post again and I'll be happy to explain:)

Let's start out by turning 7 and 1/2 into a improper fraction.

This would be 15/2. To do this, I multiplied 7 times 2, then added 1 to it. Then I put this number over 2. This is how you would turn a mixed fraction into an improper fraction. Here's a chart showing what a mixed fraction and improper fractions are:

Now that we have 15/2, we want to divide this by 1/4 to see how many 1/4-pound burgers he can make from the total ground beef he bought.

When you divide two numbers, you multiply by the recriprical. That's a pretty big word, but it just means flip-flop the second fraction.

So, 15/2 ÷ 1/4 is really

$${\frac{{\mathtt{15}}}{{\mathtt{2}}}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{4}}}{{\mathtt{1}}}}\right)$$

Now we just multiply the two fractions:

$${\frac{{\mathtt{60}}}{{\mathtt{2}}}}$$

And 60/2 is really just 30.

Omar can make 30 1/4-pound burgers out of 7 and 1/2 pounds of ground beef!

The circle has a radius of 50, so the diameter (twice the radius) is 100. Therefore, the circumference of the circle would be pi * diameter, = 100pi.

The major arc of ABC (since the points are in that order on the circle) would be the total circumference minus the minor arc of AC. Since the minor arc AC is 10pi, major arc ABC would be 100pi (circumference) - 10pi (length of AC).

If you sketch the circle, the length of the major arc and the length of the minor arc together make up the whole circumference of the circle.

Alright, here's my makeshift watch:

The red line is the rotation the minute hand makes in an hour. Note that my drawing isn't a perfect circle, but I believe the question is assuming that the rotation the minute hand makes is a full rotation.

We really just need to find the circumfrence of the red line, because that will tell us the distance that the tip of the minute hand travles in an hour.

We have the radius of this circle, and we need to find the circumference of the circle, but how do we do that?

We multiply the diameter of the circle by pi, and the diameter is 2r.

Our radius is 1.5, so our diameter is 3. (1.5*2)

$${\mathtt{3}}{cm}{\mathtt{\,\times\,}}{\mathtt{\pi}} = {\mathtt{9.424\: \!777\: \!960\: \!769\: \!379\: \!7}}{cm}$$

or rounded to the nearest tenth, 9.4 cm