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The possible factors of  7x² + nx - 11  are:

(7x - 1)(x + 11)

(7x +1)(x - 11)

(7x + 11)(x - 1)

(7x - 11)(x + 1)

Multiply out each of these and you will have all the possible values of n.

First choice

second choice is a capacitor

third choice is a potentiometer (technically a variable resistor.....like a volume control)

fourth choice is a power supply or battey

1)  Direct proof of:  If n is odd, then 4 divides (n² - 1).

      If n is odd, then there exists an integer k such that  2k + 1  =  n.

      Then:  n² - 1  =  (2k + 1)² - 1  =  (4k² + 4k + 1) - 1  =  4k² + 4k  =  4(k² + k)

                  and  4(k² + k) / 4  =  k² + k

       Also;  since k is an integer, k² + k is an integer; therefore the proof is complete.

2)  Contrapositive of:  "if n is odd, then 4 divides (n² - 1)" is "if 4 divides (n² - 1), then n is odd".

3)  Direct proof of:  If n is even, then 4 doesn't divide n² -1.

     If n is even, then there exists an integer k such that 2k = n.

     Then:  n² - 1  =  (2k)² - 1 =  4k² - 1

                 and   (4k² - 1) / 4  =  4k²/4 - 1/4  =  k² - 1/4

      estblishing a remainder of 1/4 (or, as an integer, either -3 or +3).

4)  Contrapositive of:  "if n is even, then 4 doesn't divide n² - 1"  is " if 4 doesn't divide n² - 1, then n is even".

5)  4 divides n² - 1 if and only if n is odd.

Each outfit can be paired with 4 choices of shoe

4  x 3 = 12 different pairings  (as long as her shoes match!  Haha)

i forgot to include what i thought it was! but i did it correctly. thank you. if you may help me with a few more questions.

After 28 years there will 1/2  of  10gm left =   5 gm

    after ANOTHER 28 YEARS there will be 1/2  of 5 gm left =  2.5 gm

.5 = 1 e^-k 28

k =.0247553

? = 10 e^-.0247553 (56)

? = 2.5 gm   

R2 and R3  are in parallel....their effective resistance is    (r2 * r3)/ (r2+r3) = 25/10 = 2.5 ohms

  (they each have the same current through them)

the resistance of the entire circuit is   R4 + 2.5 ohms + R1 = 12.5 ohms

The current in the circuit        V/R = I        12 v/ 12.5 ohms = 960 mA

R2 and R3 split this current    960/2 = 480 mA   (or  .48 A )

You like 2 of 13 songs.

a)  Getting both of the songs (no repeats):  (2/13) x (1/12)  =

b)  Getting neither song (no repeats):  (11/13) x (10/12)  =

c)  Getting one of them, in either order (no repeats):  (2/13) x (11/12)  +  (11/13) x (2/12)  = 

a)  Getting both of the songs (repeats allowed):  (2/13) x (2/13)  =

b)  Getting neither song (repeats allowed):  (11/13) x (11/13)  =

c)  Getting one of them, in either order (repeats allowed):  (2/13) x (11/12)  +  (11/13) x (2/12)  = 

No, it's 240

Use the Pythagorean Theorem:  d²  =  4² + 3²     --->      d²  =  16 + 9      --->     d²  =  25     --->     d  =  5 m

If the radius of the sphere is r then the length of a side of the cube is 2r.

Volume of the cube:  (2r)³  =  8r³.

Volume of the sphere:  (4/3)·pi·r³.

Portion:  (4/3)·pi·r³  /  (8r³)  =   pi/6

Find the midpoint between the two numbers:  (-7 + 5)/2  =  -1

Find the distance from the midpoint to eithr endpoint:  6

Since you want < the smaller and > the larger:  |x - -1|  >  6     or     |x + 1|  >  6

what makes you think you're so important?

volume of pyramid   =   (1/3) * (area of base) * (height)

The base of this pyramid is a 10 in by 10 in square.

area of base   =   10 in  *  10 in   =   100 in²

To find the height, let's look at this triangle:

By the Pythagorean theorem,

h² + (5 in)²   =   (13 in)²

h²   =   (13 in)²  -  (5 in)²

h²   =   169 in²  -  25 in²

h²   =   144 in²

h   =   12 in

So the height of the pyramid is  12 in.

Now we can find the volume of the pyramid.

volume of pyramid   =   (1/3) * (area of base) * (height)

volume of pyramid   =   (1/3) * ( 100 in² ) * ( 12 in )

volume of pyramid   =   400 in³

All you have to do is to sum first 7-8 terms and you will end up with 80 for the last 2 digits.

If you allow repeats of first 3 digits such as 55512, then you have =12,500 such numbers.

If repeats are not allowed for the first 3 digits such as 56712, you have =6,000 such numbers.

The expression 2z^2+13z+21 can be written as (z+a)(2z+b), where a and bare integers. What is 2a+b? 

The standard construction is ax² + bx + c  so in the example, a is 2, b is 13, and c is 21 

Going by this, (2 • 2) + 13 = 17 but I don't think that's what the question is asking for.

Actually, 2z² + 13z + 21 factors to (z + 3)(2z + 7) 

so I guess you are looking for (2 • 3) + 7  = 13 

_.

So my answer is confirmed as correct?

continued here

the intermediate value theorem says that if f is continuous on the interval [a,b] and k is in between f(a) and f(b), then there exists some value c, such that f(c) = k
 

in other words, if the function is continous on some interval, the function must pass through some value in between the values of the endpoints of the interval. 
 

since the function is a polynomial, the function is continuous everywhere. 

find f(-2) and f(4), the endpoints of the interval
f(-2) = 30
f(4) = -120

10 is in between 30 and -120, so there is some x value in between x=-2 and x=4 that makes the function value equal to 10.

to find that, set 10 equal to f(x)
\(10=-2x^3+x^2-3x+4\\ -2x^3+x^2-3x-6=0\\ x=-1\)
(i used a calculator to find x lmao)

so, when c = -1, f(c) = 10
 

a direct substitution gives you 0/0, which means more work can be done

let's simplify the numerator by subtracting the two fractions and see what we get

\(lim_{x\rightarrow 3}\frac{\frac{1}{6}-\frac{1}{9-x}}{x-3}\\ =lim_{x\rightarrow 3}\frac{\frac{9-x}{6(9-x)}-\frac{6}{6(9-x)}}{x-3}\\ =lim_{x\rightarrow 3}\frac{\frac{9-x-6}{6(9-x)}}{x-3}\\ =lim_{x\rightarrow 3}\frac{\frac{-x+3}{6(9-x)}}{x-3}\)

all i did was subtract the two fractions by giving them both common denominators. in this case, i multiplied the numerator and denominator of 1/6 by 9-x and the numerator and denominator of 1/9-x by 6. This gives the two fractions a common denominator of 6(9-x).

Let's keep going
\(=lim_{x\rightarrow 3}\frac{\frac{-(x-3)}{6(9-x)}}{x-3}\\ =lim_{x\rightarrow 3}\frac{-(x-3)}{x-3}*\frac{1}{6(9-x)}\\ =lim_{x\rightarrow 3}{-1}*\frac{1}{6(9-x)}=-1*\frac{1}{6(9-3)}=-\frac{1}{36}\)

i factored out the negative from -x+3 and moved away 6(9-x) to make it more noticeable that x-3 can be canceled out. this allows for direct substitution of x=3 which gives us -1/36

Set up a ratio

16 cm is to 128 cm²      as     24 cm is to ? cm²

16/128  =  24 /?       solve for ?

Inversely proportional    means    a = k/b       (directly proprtional would be  a = kb)

4 = k 9    results in k =  4/9

a = 4/9 b     put in b = -72   then calc 'a'

a = 4/9 (-72)       results in a = -32