geno3141

To write     x² - 6x + 3     into the form     (x + a)² + b:

Complete the square of the part  x² - 6x  by adding 9     --->     x² - 6x + 9  =  (x - 3)²

However, you cannot change the value of the expression, so if you add 9, you must also subtract 9

--->      (x² - 6x) + 3     --->      (x² - 6x + 9) - 9 + 3      --->     (x² - 6x + 9) - 6     --->     (x - 3)² - 6

So:   x² - 6x + 3   becomes   (x - 3)² - 6

This means that in   (x + a)² + b   the 'a' has value -3   and  the 'b' has value -6.

Note:  to complete the square of  x² - 6x  divide the coefficient of the 'x' term by 2 and square that value.

-6 / 2  =  -3   and  (-3)²  =  9

Since the top and the bottom of the cylinder are circles, you can use the formula:  A  =  pi · r²  to find those areas.

The area of the side can be found by using the formula:  A  =  2 · pi · r · h.

The total area will be the sum of the three answers (top + bottom + side).

Top  =  pi · 3²  =  9pi

Bottom  =  pi · 3²  =  9pi

Side  =  2 · pi · 3 · 4 =  24pi

Total  =  9pi + 9pi + 24pi  =  42pi

If you have a circle that contains two intersecting secants 

and if the point of intersection divides one secant into two parts whose lengths are 'a' and 'b'

and if that point divides the other secant into two parts whose lengths are 'c' and 'd',

then the formula:  a · b  =  c · d 

can be used.

In both problems, factor each one ...

x² - 3x - 10  =  0

(x - 5)(x + 2)  =  0

x - 5  =  0     or     x + 2  =  0

x  =  5          or     x  =  -2

2s² - 3x - 2  =  0

(2s + 1)(s - 2)  =  0

2s + 1  =  0     or     s - 2  =  0

2s  =  -1          or     s  =  2

s  =  -1/2         or     s  =  2

8 slips of paper each has a 2 on them; 2 slips of paper each has a 4 of them

The total value of all the slips when added together is (8)(2) + (2)(4)  =  16 + 8  =  24

The expected value of one draw is the average value of these 10 slips of paper is  24 / 10  =  2.4

If we add one additional 4, then the total becomes 24 + 4  =  28.

Since there are now 11 slips of paper, the expected value (average) is  28 / 11  =  2.5454...

Adding another 4, the total becomes  28 + 4  =  32.

Since there are now 12 slips of paper, the expected value (average) is  32 / 12  =  2.6666...

How many 4s  must be added to get an average of 3?

Starting with the original 10 slips of paper and the original sum of 24:

let  n  represent the number of slips of paper (each containing a 4) that we have to add to the original 10.

The sum then becomes:  24 + 4n

The number of slips of paper becomes 10 + n.

The average becomes the sum divided by the number of slips of paper:  (24 + 4n) / (10 + n).

Since we want the average to be 3, we can create this equation:  (24 + 4n) / (10 + n)  =  3

Solving this by multiplying both sides by (10 + n)     --->     24 + 4n  =  3(10 + n)

                                                                                 --->     24 + 4n  =  30 + 3n

                                                                                 --->               n  =  6

We would have to add 6 slips to the original 10.

How many  4s  must be added to get an average of at least 3.5.

Using the technique of the previous problem, we get the equation:  (24 + 4n) / (10 + n)  >  3.5

Can you take it from here?

The formula for the sum of the terms is:  S  =  n(a + l) / 2

The formula for the n^th (or last) term is:    l  =  a + (n - 1)d

Since you want to have a formula for the common difference that does not include the variable  n,  lets solve the top equation for n and then substitute that value into the second equation.

S  =  n(a + l)/2     --->     2S  =  n(a + l)     --->     2S / (a + l)  =  n

l  =  a + (n - 1)d     --->     l - a  =  (n - 1)d     --->     l - a  =  ( [ 2S / (a + l) ] - 1 )d

Solve the second equation for d:    (l - a) / ( [ 2S / (a + l) ] - 1 )   =   d

Rewrite   [ 2S / (a + l) ] - 1  with the common denominator of (a + l)     (where  1  becomes  (a + l) / (a + l) )

     --->     [ 2S / (a + l) ] - (a + l) / (a + l)

Factor out the denominator of (a + l)

     --->     [ 2s - (a + l) ] / (a + l)

Look back at the left-hand side of the equation for  d  and replace  ( [ 2S / (a + l) ] - 1 )  with   [ 2s - (a + l) ] / (a + l):

     --->     (l - a) / ( [ 2S / (a + l) ] - 1 )   =   (l - a) / { [ 2s - (a + l) ] / (a + l) }

Multiply both the numerator and denominator of this expression by (a + l):

     --->     (l - a)(a + l)  /  { [ 2s - (a + l) ] / (a + l) · (a + l) }      (the (a + l)s on the right will cancel)

     --->     (l - a)(a + l)  /  [ 2s - (a + l) ]

Simplifying and interchanging, this becomes:  ( l² - a² ) / [ 2s - (l + a) ]   which equals  d.

In the right triangle formed by the floor, the rise, and the ramp, the rise is the opposite side and the ramp is the hypotenuse.

sin(angle)  =  opposite side / hypotenuse     --->     sin(angle)  =  0.7 / 4.2     --->     sin(angle)  =  0.166667

--->     angle  =  sin^-1(0.166667)  =  9.6^o  (approximately)

Volume(prism)  =  Area(base) x height

Volume  =  6x · 3.5x

3024  =  21 x²

144  =  x²

--->     x  =  12 inches

The proportion of the variation that is explained is  r².

Therefore, in this case, since  r = 0.400,  r²  =  0.160,  so the correct answer is  16%.

One way to solve  5 + 12 / x  =  -1

is to first subtract 5 from both sides:     --->     12 / x  =  -6

then, multiply both sides by x:               --->     12  =  -6x

Finally, divide both sides by -6:             --->     -2  =  x