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Using the quadratic formula, the difference  in the  roots will simplify to  sqrt (k^2 -20)

So....we have

sqrt ( k^2 - 20) =  2sqrt (31)

k^2 - 20  = 4 ( 31 )

k^2  = 124  + 20

k^2  = 144

k = 12, -12 

If  k  = 12     we have

x^2 + 12x  + 5  = 0

And   sqrt (12^2 - 20) =  sqrt (124) =  sqrt ( 31 * 4) =  2sqrt (31)

So 12 is the largest possible k

Powers of 13 (starting with 13^1) end in

3, 9, 7, 1    a cycle of  4

So

13^19  ends in 7

Powers of 19  (starting with 19^1) end in

9, 1      a cycle of 2

So

19^13  ends in 9

So

13^19 * 19^13  ends in

7  * 9  =     ...3

10, 15, 20, 25, 30  = 100

Sorry, I'm a little lost. There are multiple points in your answer which confuse me. 
Why is XY = XZ = radius (I'm assuming R is radius) because X lies on the perpendicular bisectors of all three sides of the triangle? Points Y and Z lie on a circle centered on O, so OY and OZ would be the radius, unless you're referring to another circle? 

What is the inscribed circle? 

How did you get 48/(2 * sqrt(3)) = 12sqrt(3)?

Sorry, and thanks.

y =ax^2 + bx  - 6

a^2 = 4

If this lies  completely below the   x axis, then a must  be negative because the  parabola will  turn  downward

So....a^2 =4  and  a  = -2

Since this lies  below the  x axis it  will  have no  real  roots....therefore, the discriminant  must  be < 0

b^2 - 4(-2)(-6)  < 0 

b^2  -48 < 0

b^2 < 48

b < 7  →     the largest value of b  =  6

Simplify the polynomial as

x^2  + 6x + 5

This can be factored as   (x+5) (x +1)

A,B  = 1

AB = (1)(1)  =  1

To factor the given expression $6x^2 + 17x + 5 - 5x^2 - 11x$, let's combine the like terms first:

$6x^2 + 17x + 5 - 5x^2 - 11x = (6x^2 - 5x^2) + (17x - 11x) + 5 = x^2 + 6x + 5$

Now we need to factor the expression $x^2 + 6x + 5$ into the form $(Ax+1)(Bx+5)$. We'll look for integers $A$ and $B$ that satisfy this factorization.

To do this, we need to find two numbers that multiply to the coefficient of the constant term (which is 5) and add up to the coefficient of the linear term (which is 6). These numbers are 5 and 1, because:

$5 \times 1 = 5$ (constant term) $5 + 1 = 6$ (linear term)

So, we can factor $x^2 + 6x + 5$ as:

$x^2 + 6x + 5 = (x + 5)(x + 1)$

Thus, $A = 1$ and $B = 5$.

The value of $AB$ is $A \times B = 1 \times 5 = 5$.

Let's start by finding the radius of the circle. Given that the area of the circle is 48pi, we can write:

Area of the circle = pi * r^2
48pi = pi * r^2

Dividing both sides by pi:

48 = r^2

Taking the square root of both sides:

r = sqrt(48)

Since triangle XYZ is equilateral and X is the circumcenter, we know that X is equidistant from Y and Z. Let's call this distance d. In other words, d = XY = XZ.

Now, let's find OD (the distance from O to D), where D lies on YZ and XD ⊥ YZ. Since triangle OYZ is a right-angled triangle (it's not stated but assumed), we can use Pythagorean theorem:

OD^2 + XD^2 = r^2

Substituting r with sqrt(48):

OD^2 + XD^2 = 48

In an equilateral triangle, the median also acts as an altitude and bisects the opposite side. So OD is half the length of side YZ.

Now let's call s as a side of triangle XYZ:

OD = (s/2)

XD can also be represented as (YZ/2) or (s/2).

Plugging these values into our previous equation:

((s/2))^2 + ((s/2))^2 = 48

Simplifying:
(s^2)/4 + (s^2)/4

Combining like terms:

(s^2)/2 = 48

Multiplying both sides by 2:

s^2 = 96

Taking square root of both sides:
s = sqrt(96)

Now that we have found the side of triangle XYZ, we can find its area using Heron's formula or by simply using this formula for equilateral triangles:

Area = (s^2 * sqrt(3))/4

Substituting the value of s:

Area = (96 * sqrt(3))/4

Area = 24sqrt(3)

So, the area of triangle XYZ is approximately 24sqrt(3).

Probability of rolling a '1'     is   1   out of  ( 1+2+3+4+5+6)  = 1 out of 21

    probability of rolling TWO '1s'  is    1/21 * 1/21 = 1/441 

a has to be 2, 4, or 6

and b has to be 2,4, or 6

so yes,  0.5*0.5 = 0.25

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GA

--. .-   

This is for the first problem.

We can start by cross-multiplying the two expressions:

(2x^2 + x + 3)(x + 1) = (2x + 1)(x^2 + x + 1)

Expanding both sides, we get:

2x^3 + 3x^2 + 3x + 2x^2 + 2x + 3 = 2x^3 + 3x^2 + 2x + x^2 + x + 1

Combining like terms, we get:

5x^2 + 5x + 4 = 3x^3 + 4x^2 + 3x + 1

Moving all the terms to the left-hand side, we get:

3x^3 - 5x^2 - 2x - 3 = 0

We can factor this expression as:

(x - 1)(3x^2 + 4x + 3) = 0

The first factor, x−1, will always be nonzero, so the only solution is x=1​.

The units digit for powers of three is: 3,9,7,1,3... The 19th number will be 7.

The pattern for 9 is: 9,1,9... and the thirteenth number will be 9

Then we have 7*9, which has a units digit of 3

A piano has many keys, but it cannot open a single door because it is a musical instrument and not a lock or a key.

The height of  one  of these triangles = (1/2) diagonal of the square =  (1/2) 10 *sqrt (2) =  5sqrt 2

Half the base length =  5 sqrt (2) / sqrt (3) ) ...so the whole base length = 10sqrt (2) / sqrt (3)

Total area of both triangles =  2 (1/2) ( 5 sqrt (2)) (10 sqrt (2) / sqrt (3) = 

50 * 2 / sqrt (3)  = 

100 /sqrt (3) ≈ 57.735

1 / 4 because 1 / 2 * 1 / 2 = 1 / 4

We have a binomial probabiity  problem....3 of the  dice  must  show odd  and each one has a probability of (1/2) of that....the other 7 each have a probability of (1/2) of showing even.....we need to  choose any  3 of the  10 to show  odd...so we have

C(10,3) (1/2)^3 (1/2)^7  =  15 /128 ≈ .117

Let's set up a system of equations based on the given information:

Let "x" be the number of stamps Jeremy initially has.

According to the first condition, if he gives each friend 2 stamps, he will have 6 stamps left: x - 2f = 6, where "f" is the number of friends.

According to the second condition, if he gives each friend 3 stamps, he will be short of 1 stamp: x - 3f = -1.

Now have a system of two equations:
x - 2f = 6

x - 3f = -1

Let's solve this system to find the value of "x" (the number of stamps Jeremy has):

Subtracting equation 2 from equation 1 gives: 
(x - 2f) - (x - 3f) = 6 - (-1)

x - 2f - x + 3f = 6 + 1

f = 7.

Now can substitute the value of "f" into either equation 1 or 2 to solve for "x." Let's use equation 1: 
x - 2(7) = 6

x - 14 = 6

x = 6 + 14

x = 20.

Therefore, Jeremy has 20 stamps.

So, the correct answer is:

1. 20

AB = 1

The interior angle of the  octagon = 135

The exterior angle PAB  =360 / 8  =  45

Angle ABP  = (360 -135) / 2  = 112.5

Angle APB = 180- 45 -112.5 = 22.5

Using the Law of Sines

PB / sin 45 = AB / sin 22.5

(sin 22.5) = sqrt [ ( 1 -sin 45 ) / 2  ]  =   sqrt [ (1 - sqrt (2)/2) /2 ]  =  sqrt [( 2 - sqrt (2)) / 4 ] = sqrt [ 2 -sqrt (2)] / 2  

PB / (1/sqrt (2)) = 1  / ( sqrt [2 -sqrt (2) ] / 2) =   

PB  =  2 ( 1/sqrt (2)) /  sqrt [2 -sqrt (2) ]  =  sqrt (2) / sqrt [ 2 -sqrt (2) ]  =

sqrt [(2 ) * sqrt( 2 + sqrt(2)] / sqrt [ 4 -2 ] =  sqrt (2) * sqrt (2 + sqrt (2) / sqrt (2)=

sqrt [ 2 + sqrt (2)]