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The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Now let's substitute the values into the formula:

s = (-(-6000) ± √((-6000)^2 - 4*11*(-27500))) / (2*11)

Simplifying further:

s = (6000 ± √(36000000 + 1210000)) / 22
s = (6000 ± √(37210000)) / 22
s = (6000 ± 7200) / 22

We need s to be positive, so s = (6000 + 7200)/22 = 600.

The answer is 600.

We are looking for the two points where the equations intersect....where they are equal ...so equate the two circle equations 

x^2 + y^2 -4     =  ( x-5)^2 + (y-8)^2 - 57            expand  ...and then simplify to 

0 = -10x -16y+36    which rearranges to 

y = - 10/16 +36/16 

y = - 5/8 x +9/4      this is the  equation of the line AB......the slope is - 5/8 

Here is the DESMOS graph showing the two circles and the (green) line:

Unfortunately, your post does not show the 20 x 20 matrix. I have constructed a 20 x 20 matrix reading your data from left to right thus:

From a simple spreadsheet, it can be determined that the greatest number resulting from the product of any consecutive 4 numbers either vertically or horizontally without wrapping columns or rows is 51,267,216. The smallest number is of course 0. The median number is 1,944,852, and the average is 4,256,955 (rounded). There are 578 computations, and the sum of those computations is 2,460,519,837.

Hope this helps.

By the Pythagorean Theorem, PQ2+PR2=QR2, so 52+82=112.

Thus △PQR is a right triangle where PQ is the hypotenuse, so by the Pythagorean Theorem, PM2=(2PR​)2=(28​)2=16.

Hence, PM=16​=4​.

By the Angle Bisector Theorem, [ \frac{AZ}{ZB} = \frac{AC}{BC} = \frac{CY}{BY} = \frac{8}{14-Y}. ] Since AY+Y=14, [\frac{AZ}{ZB} = \frac{8}{14-AY} = 8.] Let ZB=x. Then AZ=8x. Since AZ+ZB=14, x+8x=14, so x=2. Therefore, BZ=2.

Let CZ=y. Then CY=8y. Since CY+YZ=14, y+8y=214​=7. Therefore, [BC = CY + BZ = 8y + 2 = 8 \cdot 7 + 2 = \boxed{13.0909}.]

By the Angle Bisector Theorem, [ \frac{AZ}{ZB} = \frac{AC}{BC} = \frac{CY}{BY} = \frac{8}{14-Y}. ] Since AY+Y=14, [ \frac{AZ}{ZB} = \frac{8}{14-AY} = 8. ] Let ZB=x. Then AZ=8x. Since AZ+ZB=14, x+8x=14, so x=2.

Therefore, BZ=2​.

Sure, here's how to find the slope of line AB:

Rewrite each circle equation in standard form:

x^2 + y^2 = 4 => (x^2 - 4) + y^2 = 0

(x - 5)^2 + (y - 8)^2 = 57 => (x^2 - 10x + 25) + (y^2 - 16y + 64) = 0

Add the two equations: 2x^2 - 10x + y^2 - 16y = 39

Group like terms: 2(x^2 - 5x) + y^2 - 16y = 39

Factor out the common factors: 2(x - 5)(x - 5) + y(y - 16) = 39

Solve for y: y = (39 - 2(x - 5)(x - 5)) / (y - 16)

To find the slope of line AB, we need to find the coordinates of points A and B. We can do this by substituting different values of x into the equation for y and solving. For example, if we substitute x = 5, we get y = 11. Therefore, point A is (5, 11).

Similarly, if we substitute x = 8, we get y = 0. Therefore, point B is (8, 0).

Calculate the slope of line AB: (11 - 0) / (5 - 8) = 2

Therefore, the slope of line AB is 2.

To find a linear inequality with the given solution set, we need to consider the shaded region. Let's denote the coordinates of the shaded region's vertices.

Looking at the image, we can see that the shaded region is below and to the left of the line passing through the points (4, 1) and (5, 3). We can use these two points to find the equation of the line in slope-intercept form (y = mx + b):

First, find the slope (m): m=5−43−1​=2

Now, we can use the slope and one of the points (let's use (4, 1)) to find the y-intercept (b): 2⋅47b=−7

So, the equation of the line is y=2x−7.

Now, since we want the shaded region to be below and to the left of this line, we want the region where y<2x−7.

We can rewrite this inequality in standard form (ax + by + c > 0) by moving all terms to one side

−2x+y+7<0

So, the linear inequality that represents the shaded region in standard form is −2x+y+7<0.

Case 1: The number has an even number of digits.

Since the sum of the digits is 12, this is impossible.

Case 2: The number has an odd number of digits, and the units digit is 1.

In this case, the first digit must be 3, and the digits in between must add to 9. There are (25​)=10 ways to choose the two numbers that add to 9, and 22=4 ways to order them. Thus the number of possibilities in this case is 10⋅4=40.

Case 3: The number has an odd number of digits, and the units digit is 3.

In this case, the first digit must be 1, and the digits in between must add to 3. There are (13​)=3 ways to choose the number that adds to 3, and 2 ways to place it. Thus the number of possibilities in this case is 3⋅2=6.

The total number of possibilities is 40+6=46​.

From the definition, [ a \mathbin{\spadesuit} (a + 1) = \frac{a^2}{a+1} = 9. ] Multiplying both sides by a+1 yields a2=9(a+1). Substituting a+1 for x and 9 for y into the definition of ♠, we find that this simplifies to 9(a+1)2​=9, which leads to a+1=±3. Therefore, a=−2,2​.

Let f be the probability that a freshman is elected. Then 4f is the probability that a sophomore is elected. Since the sum of the probabilities must be 1, we have f + 4f = 1, which implies f = 1/5. Thus 4f = 4/5. The fraction of the swim team that are sophomores is equal to the probability that a sophomore is elected, which is 4/5. So the answer is 4/5.

To find the coordinates of the center of the circle, we can use the midpoint formula. The midpoint formula states that the midpoint of a line segment with endpoints (a,b) and (c,d) is the point (2a+c​,2b+d​).

Using the midpoint formula, we can find the midpoints of the following line segments:

The midpoint of the line segment connecting (22,15) and (−25,0) is (222+(−25)​,215+0​)=(−1.5,7.5).

The midpoint of the line segment connecting$(-25,0)$ and (22,−29) is (2−25+22​,20+(−29)​)=(−1.5,−14.5).

The midpoint of the line segment connecting (22,−29) and (22,15) is (222+22​,2−29+15​)=(22,−7).

Since the three midpoints coincide, we can conclude that the center of the circle is the point (−1.5,−7)​.