All Answers 356000 Answers

2 people: 8 choose 2 = 28 ways

3 people: 8 choose 3 = 56 ways

4 people: 8 choose 4 = 70 ways

5 people: 8 choose 5 = 56 ways

6 people: 8 choose 6 = 28 ways

7 people: 8 choose 7 = 8 ways

8 people: 8 choose 8 = 1 way

Add all the possibilities up, and you get 247.

160 x 1.03 = $164.80

Ah! Thanks, proyaop! (and Tiggsy) That makes sense. Thanks for all your help!

First we open parenthesis.

\(w^2 + 26w + 169 = (3w + 7)(2w + 5)\)

Then we open parenthesis on the right side of the equation.

\(w^2 + 26w + 169 = 6w^2 +29w + 35\)

Then we have to combine like terms.

\(5w^2 + 3w -134 = 0\)

Now we can use the quadratic formula: w = \(-b {+\over} \sqrt{b^2 - 4ac}\over2a\) where a is the coefficient of \(w^2\), b is the coefficient of w, and c is the constant.

Now to plug in the values into the formula: we get

w = \(-3 {+\over}\sqrt{2689}\over10\)

Web2.0calc does not allow AoPS questions.  Your question was automatically flagged.

sorry, but... its wrong!

By quadratic formula, b = 8.

By geometric series, 1 - x + x^2 - x^3 + ... = 1/(1 + x).

The equation is then x = 1/(x + 1).

Then x^2 + x - 1 = 0.

By Vieta's formulas, the sum of the roots is -1.

By the Binomial Theorem, the constant term in the expansion of $(2z - 1/\sqrt{z})^9$ is 1344.

Again by the Biniomal Theorem, the constant term in the expansion of $(x^2 + 1/x)^4$ is 6.

I've burned piles of HUNDREDS of tumbleweeds......they burn almost explosively !

anyway     he burned 2/3 s of them    2/3 * 30 = 20 burned

GCM of 12 and 18   is   six   bags of 2 peanut and 3 oatmeal

Let the total number of sweets==C
Let the share of each boy and each girl=s

C/27==s.....................................(1)                
18s - 1/4*(18s) ==9 [s +12].........(2), solve for s, C

Use substitution to get:
C==648 - total number of sweets.
s==24 sweets - originally shared by 18 boys and 9 girls each.

There would be 8 options for the first day, 7 for the second, 6 for the third, and so on... So 8x7x6x5x4x3x2. However, since the fruits are indistinguishable, we have to divide by 3!x2!x2!.

This gives us: \({{8*7*6*5*4*3*2} \over 3!2!2! } = 420\) ways.

Add up the amount of fabric:

\(2 {1 \over2} + 1 {3 \over 4} = 4 {1 \over4}\)

Subtract the total amount from 3:

\(4 {1 \over4} - 3 = 1 {1\over 4}\)

This means that she needs an extra \(1 {1\over 4}\) yards of fabric.

To find the total cost, do \(1 {1\over 4} * $7.98 = $9.98\)

Options a and b are eliminated because we don't know anything about f(x) when x is in range -1 < x < 2.

Looking at c and d, we try to see which one makes sense. Because we know in this range of -2 < x < 1, f(x) will be negative, we can rewrite c and d.

c) = |f(x)| = f(x) because the -1 * -1 = 1

d) |f(x)| = -f(x) since f(x) in range -2 < x < 1 is negative.

Therefore the answer is c).

If you consider the "squares" as rectangles, then we can count the number of rectangle sizes there are.

For the size 1x1 there are 12 squares/rectangles.

For the next size which is a 2 x 1, there are 17 rectangles.

For the next size which is a 3 x 1, there are 10 of those rectangles.

For the next size which is a 4 x 1, there are 3 of those rectangles.

For the next size which is a 2 x 2, there are 6 of those squares/rectangles.

For the next size which is 2 x 3, there are 7 of those rectangles.

For the next size which is 2 x 4, there are 2 of those rectangles.

For the next size which is 3 x 4, there is 1 rectangle.

For the last size which is 3 x 3, there are 2 of those squares/rectangles.

Altogether there are 40 rectangles and 20 squares. (You might count squares as rectangles.)

Count it by shape:

length x width

1 x 1: 12 

1 x 2: 9

1 x 3: 6

1 x 4: 3

2 x 1: 8

2 x 2: 6

2 x 3: 4

2 x 4: 2

3 x 1: 4

3 x 2: 3

3 x 3: 2

3 x 4: 1

Add everything up, and you get 60.

Its just the multiples of 3 from the smallest possible sum to the largest. 

From 3 + 6 + 9 = 18, to 36 + 39 + 42 = 117.

This is from the 6th multiple of 3, to the 39th multiple of 3.

6 - 5 = 1, 39 - 5 = 34

There are 34 distinct sums.

This logic question has been answered here already:

2x(x + 10) = -50

2x^2 + 20x = -50

2x^2 + 20x + 50 = 0

Using VIeta's formulas, the sum of roots of the quadratic ax^2 + bx + c = 0 is -b/a.

-b/a = -20/2 = -10. 

(As ElectricPavlov has solved this question, the two roots are -5 and -5, which sum to -10 as expected.)

2x(x + 10) = -50     expand and simplify to

2x^2 + 20x + 50 = 0   or

x^2 + 10x + 25 =0

(x+5)(x+5) = 0     shows only one root x = - 5    (a 'double root')

     sum = -5     

fabric required: 2 1/2 + 1 3/4 = 3 + 1/2 + 3/4 = 4 1/4 yards

fabric in possession: 3 yards

============================

need to purchase 4 1/4 - 3 = 1 1/4 yards

total cost = (1 1/4) * ($7.98) = $9.98 (to the nearest cent)