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First, let's combine all like terms on the right side of the equation. 

We get the formula

\(y=-5x^{2}-6x+10\)

Now, it's important to note that the x coordinate of the vertex can be written in the form

\(\frac{-b}{2a}\) for parabolas in the form of \(y=ax^2+bx+c\)

Thus plugging in -5 and -6, we get

\(x=6/-10 = -3/5=-0.6\)

Plugging this back into the equation, we fins that \(y=11.8\)

Thus the vertex is at \((-0.6, 11.8)\)

So our answer is \((-0.6, 11.8)\)

Thanks! :)

First let's combine all like terms and simplify. We have'

\(x^{2}+18x+3\)

Now, we complete the square for x. Since it is 18x, we add and subtract 81 from the equation. We have

\(x^2+18x+81+3x-81\\ (x+9)^2-78\)

Thus, the first blank is 1, the second blank is 9, and the third blank is -78. 

Thanks! :)

For n=0,1,2,3,4 determine the pattern of the powers of n mod 5:

n=0: n to any power is 0; the remainder when divided by 5 is always 0
n=1: n to any power is 1; the remainder when divided by 5 is always 1
n=2: the pattern is 2, 4 (=-1), -2, 1 (cycle length 4)
n=3: the pattern is 3, 9 (=-1), -3, 1 (cycle length 4)
n=4: the pattern is 4 (=-1), 1 (cycle length 2)

f(n) is equal to (n^8+n^16) mod 5.


The lengths of the cycles of the remainders are 1, or 2, or 4; 8 and 16 are both multiples of all of those. So both n^8 mod 5 and n^16 mod 5 will be the last number in the pattern for each value of n.

The last numbers in the patterns are 0 for n=0 and 1 for all other values of n. So

f(0) = 0+0 = 0
f(1) = 1+1 = 2
f(2) = 1+1 = 2
f(3) = 1+1 = 2
f(4) = 1+1 = 2

f(0)+f(1)+f(2)+f(3)+f(4) = 0+2+2+2+2 = 8

ANSWER: 8

Hope it Correct!

(x,y) = (3 + 2*sqrt(2), 7 - 2*sqrt(2))

The minimum value is 5/9.

We put the system in matrix form:

[2 1 0] [t] = 3.15

[0 2 1] [b] = 4.85

[1 2 1] [m] = 5.25

Solving, we get b = 1.40, t = 1.20, m = 1.60.  Therefore, one bagel costs 140 cents.

Find the units digit of $13*287$.   

13⁰ = 1            13⁴ = 28561            13⁸ = 615730721               13¹² = 23298085122481   

13¹ = 13          13⁵ = 371293          13⁹ = 10604499373           13¹³ = 302875106592253   

13² = 169        13⁶ = 4826809        13¹⁰ = 137858491849       13¹⁴ = 3937376385699289   

13³ = 2197      13⁷ = 62748517      13¹¹ = 1792160394037      13¹⁵ = my calculator went into scientific notation   

The above tabulation shows that the units digit repeats every four exponents.   

Divide the exponent by 4, disregard the whole number part of the   

quotient, and consider only the remainder part of the quotient.   

Raise 13 to that remainder and see what the units digit is.   

Pick a few examples to demonstrate process.   

13⁶      6 / 4 = 1 R 2      13² = 169      therefore the units digit of 13⁶ is 9.   

13⁷      7 / 4 = 1 R 3      13³ = 2197    therefore the units digit of 13⁷ is 7.   

13¹²    12 / 4 = 3 R 0     13⁰ = 1         therefore the units digit of 13¹² is 1.   

13²⁸⁷   287 / 4 = 71 R 3    13³ = 2197    therefore the units digit of 13²⁸⁷ is 7.   

_.

a square foot is 0.3048 x 0.3048 = 0.09290304 square meters. The wall is 1440 square meters. A gallon covers 0.09290304 x 150 = 13.935456 square meters. So just divide 1440 by 13.935456, which is between 103 and 104(around 103 1/3), so the answer is 104

Or 133 and 79 are relatively prime, so 2

Catherine rolls a standard $3$-sided die six times. If the product of her rolls is $600,$ then

how many different sequences of rolls could there have been? (The order of the rolls matters.)   

I can't find on the internet, what the numbers are on a "standard" 3-sided die.  

A couple of sites said the faces are 1, 2, and 3.   

Other places said the faces have other numbers.   

The prime factors of 600 are 2, 3, and 5.  

Of course, 1 is a factor of 600, but by definition 1 is not a prime number.   

If the numbers on the die are 1, 2, and 3, you can't get there from here.   

_.

On planet Enigma, the residents use a currency called the confusion.  There are only two confusion bills on Enigma, one worth $7$ confusions and the other worth $12$ confusions.  There are also some coins of smaller value, but each weighs over $10$ kilograms, so they are difficult to carry around.  In how many ways can a resident of Enigma use only bills to purchase a toaster that costs 19 confusions?   

I think nobody is answering this problem because it's so obvious that they suspect a trick.   

Well, I'll admit I'm suspicious, too, but I'll take one for the team.  I'm going out on that limb.  

There is only one way to purchase the toaster ... with one 7-confusion bill and one 12-confusion bill.   

_.

Let's note something really important first. 

In the problem, it said that \( \overline{AB} =4\)

Then, it asks us to find the length of AB. 

So, our final answer is \( \overline{AB} =4\)

So 4 is our answer. 

Thanks! :)

First, let's convert every single measurement into feet from meters so we don't have to worry about it later.

We have \(\frac{12m}{ 0.3048 m/ft }=39.3701 ft \) for the height

And we have \(\frac{120m}{ 0.3048 m/ft }=393.7007 ft \) for the length. 

Now, we can find the total area of the wall. We do length times height, so we have

\( (39.3701 ft)(393.7007 ft) = 15,500.0359 ft^2 \)

Now, we do some ugly math. We must divide by the gallons. 

Since each gallon of paint covers 150, we must do

\(\frac{ 15,500.0359 sq ft}{ 150 sq ft/gallon }= 103.3335 gallons \)

Since you can't actually by a fraction of a gallon, we must round up. 

So, we simply have

104 gallons as our answer. 

So 104 is our answer. 

Thanks! :)

In a store window, there was a box of berries having a total weight of 200 kg.  The berries were 99% water, by weight.  After two days in the sun, the water content of the berries was only 90%, by weight.  What was the total weight of the berries after two days, in kg?   

The water was 99% of 200, so the pulp was 1% of 200 = 2 kg.   

After two days in sun, 90% is water, but the pulp didn't evaporate.   

There's still 2 kg pulp, which now accounts for 10% of the berries.   

Since 10% is 2 kg, the total weight of the berries is now 20 kg.   

_.

We can write a system of equations to solve the problem. 

Let b be bageld. 

Let t be toast.

Let m be muffins. 

We have the system

\(2T + 1B = 3.15 \\ 1M + 1B = 3.50 \\ 1T + 2B + 3M = 8.15\)

We want to find what a bagel is. So, from the first two equations, we can isolate the other variables in terms of B. 

We get

\([3.15 - B ] / 2 = T \\ M = 3.50 - B\)

Plugging this into the third and final solution, we find that

\([ 3.15 - B ] / 2 + 2B + 3 [ 3.50 - B ] = 8.15 \\ [3.15 - B ] + 4B + 6 [ 3.50 -B ] = 16.30 \\ -3B + 3.15 + 21 = 16.30 \\ -3B = 16.30 - 3.15 - 21 \\ -3B = -7.85 \\ B = 7.85 / 3 ≈ $2.62\)

This is about 262 cents or 262.67

Thanks! :)

First, we must find the slope of the two points. 

Since we have

\(\frac{150-17}{82-3} = \frac{133}{79}\)

This means for every 133 latice points to the right of 3, you move up 79. 

However, note that when we add 133 to 17, we get \(133+17=150\)

This means there are no lattice points between \((3, 17), (82, 150)\)

So 2 is our answer. 

Thanks! :)