All Answers 358352 Answers

1 / ( 1 + 1/sqrt (3)  + 2)  =

1/ ( 3 + 1/sqrt (3) )   = 

 sqrt (3)    /  ( 3sqrt (3)  + 1 )      multiply numerator /denominator by   3sqrt (3)   - 1

sqrt (3) ( 3sqrt (3)  - 1)

__________________  =

     9*3   - 1

9 - sqrt (3)

________

      26

x^2 + y^2  = 4(x + y)  + 12

x^2 + y^2  = 4x + 4y + 12

x^2 - 4x + y^2 - 4y  = 12       complete  the square on x, y

(x^2  - 4x  + 4)  +( y^2 - 4y + 4 )   =   12 + 4 + 4

( x - 2)^2  + ( y -2)^2 =   20

This is a circle centered at (2,2)  with a radius of sqrt (20)

The greatest value of  x =    2 + sqrt (20)

There are \(10^4=10,000\) ways to choose M, A, T, and H. There is only one way to choose M, A, and H and two ways to choose T in MATHCOUNTS. 

That means that there are \(1 \cdot 1 \cdot 1 \cdot 2 = 2\) ways to spell MATH. Then, the probability of getting MATH is

\(\dfrac{2}{10,000} = \boxed{\dfrac{1}{5,000}}.\)

And yes, HeWhoShallNotBeNamed (@Voldemort)s' answer was correct.

On the graph......go up to the horizontal line between 350 and 400 litres.....this is  the line representing 375 liters

Now the red line crosses this  between   80  and 90  gallons

So    375 liters  ≈    85 gallons

Let M be the mid-point of AB

Then triangle PMB  is a 30-60-90 right triangle with MB  = 3   PM = sqrt 3   and   PB = sqrt (12)  = the radius of the large circle  = R

So  the radius of one of the smaller circles =  

[ R  - PM ]  / 2  =   [  sqrt (12)  - sqrt (3) ]  / 2  =   

So......the total area of the three small circles = 

3 * pi *    ( [  sqrt (12) - sqrt (3) ] / 2 )^2  =

(3/4) pi * [ 12 - 2sqrt (36) + 3]  =

(3/4)pi  * [ 12 - 12 + 3]  =

(3/4)pi (3)  =

(9/4) pi cm^2

Rearrange the given equation as

x^2 - 2x +  8  =  0

(x - 4) ( x + 2)  = 0

The roots  are   4 , -2 =   a , b

4^4 =  256 =  a^4

(-2)^4  = 16 = b^4

The sum of these roots =  -b  in the new quadratic  = -272

The product of these roots = c in the new quadratic =   4096

The new quadratic =  x^2 - 272x + 4096

No. of three digit multiples  of 5  =

(995  - 100)/5  + 1  =    180

Number of three digit multiples of  8  =

(992 - 104) / 8  + 1  =    112

Number of three  digit multiples of 5 and 8  =  those  divisible by 40

( 960 -  120)  / 40  +  1    =   22

Number of three digit multiples  of  5 and 8  =   180 + 112  -  22   =  271

Number of three digit multiples of neither  5 or 8   =    900  - 271   =  629

Let r be the original radius

We have that

New  area -  Old area =    43 cm^2

pi (r + 1)^2   -   pi r^2 =   43

pi ( r^2  + 2r + 1)  - pi r^2  = 43

r^2 + 2r + 1  - r^2  =   43 / pi

2r + 1  =  43/pi

2r  = 43/ pi  -  1

r =  (43/ pi - 1)   / 2  ≈     6 cm

The answer is 8.2.

Let a  =  1

So

( x + 3) ( x - 8)    =  0

x^2 -5x - 24  =  0

b  = -5

[P.S. -   the value of "a" = 1  is  arbitrary.......other values of  b are possible  with other choices of "a"  ]

\($\frac14 < \frac{x}{5} < \frac23$\)

1/4 <  x /  5        and       x / 5 <  2/3

5/4 <  x                          x <  10/3

So.....the possible integer  values for x  are   2 , 3

Can you format it a little better? I tried typing the url and it didn't give me builderboi's answer. I was sure I didn't mistype it! To be sure you can reupload it please so I can copy and paste, in regular text so I can

look at builderboi's answer, not mine

\(https://web2.0calc.com/questions/counting-and-probability_7\)

First, simplify the equation to \(x^2 - 3x + 4 = 0\)

Note that \(a^3b + b^3 a = ab(a^2 + b^2) = ab((a+b)^2 - 2ab)\)

By Vieta's, \(a + b = {b \over a} = -{ -3 \over 1} = 3\) and \(ab = {c \over a} = 4\)

Can you take it from here?

The answer is 2

It can have the value of 1.....

so that is wrong.

m cannot have the values 1, 2, and 3.

The correct answer is S R P Q

2x-1 is linear. You can literally plug the least value in f(x), which is negative 8, to get a new one, -17.

Doing the same with four gives 7.

The domain of g(x) is [-17, 7].

Thank you all for your answers, I was kinda stumped when I reached CPhill's answer, and it was wrong.

TYSM

Hmm  the thing is, it may be talking about the major arc instead of the minor arc. 

Edit: Nevermind, I thought nerdiest was saying that 8pi was wrong. Sorry! :(

here

By symmetry, \(\widehat{BD}=\widehat{CA}=100^\circ\). Furthermore,\(\widehat{AB}=\widehat{CD}\) , so\([360^\circ=\widehat{AB}+\widehat{BD}+\widehat{DC}+\widehat{CA}=2\widehat{AB}+200^\circ\)Therefore the arc \(\widehat{AB}\) measures \(80^\circ\). Since the diameter of the circle is \(36''\), the length of the arc is \(\frac{80}{360}(\pi\cdot36)=\boxed{8\pi}\text{~inches}.\)

Welp, I guess both of you agree!

Here's my attempt: 

Because \(AB \parallel CD\), \(\angle DAB = \angle ADC = 50^\circ\).

Now, draw line segment \(CB\), and label the intersection point \(E\).

Note that \(\triangle ECD \) is isosceles, so \(\angle ECD = 50^ \circ\). This means that \(\angle ECD = \angle AEB = 80^ \circ\)

So, the length of \(\overset{\large\frown}{AB} = {80 \over 360} \times 2 \times 18 \times \pi = \color{brown}\boxed{8 \pi }\)