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Repeated roots occur when the quadratic can be factored into two or more binomials. 

For example....

(x^2 + 6x + 9) has repeated roots because it can be factored into (x + 3)(x + 3), which only give one distinct root which is -3. 

On the other hand...

(x^2 + 5x + 6) does not have repeated roots because it cannot be factored into a product of two of the same binomials and factors to (x + 2)(x + 3), which gives us too distinct roots which are -2, and -3. 

The first one doesn't factor evenly. 

The second one factor into 3(x-8)(x+7), which has two distinct roots which are 8, and -7. 

The third one doesn't factor evenly. 

The fourth one factors into (5x-3)(5x-3) which has repeating roots which are 3/5 and is the answer. 

The fifth one factors into (x-12)(x+2) which has two distinct roots which are 12 and -2. 

Therefore, our answer is the 4th quadratic which is 25x^2 - 30x + 9, which factorizes (5x-3)^2 and has one repeating root that is 3/5. 

floccinaucini, if you didn't know, when written in the form ax^2+bx+c, the discriminant of the quadratic equation is b^2-4ac.

Hope it helps!

Thanks, HELPMEEEEEEEEEEEEE    !!!!

Here  is  another  method

If the discriminant  = 0   we  have a repeated root

(1)  (18)^2 - 4(-1)(81)  > 0      NO

(2)  (-3)^2  - 4(3)(-168) > 0    NO

(3)  (-4)^2 -  4(1) (-4) >  0    NO

(4)  (30)^2 -4(25)(9) =  900 - 900  =  0    [ this is the one with the repeated root ]

That's good. 

Yeah, I knew that but accidentally misread the question, but I decided to post both for the viewer's knowledge. 

Again, we need the measure of the radius to find the length of the arc. I'm not sure if the 8's are part of your question, but if so, the arc length is 4*3.14= 12.56. If the 8's are not part of your question, you need to find the radius of the circle that the problem gave you. Finding the arc length was already explaing twice now.

Hope it helps!

Or... Another way to solve it is this, we can make an equation. 

We know that the cheetah population is decreasing by 10% each year so each year it will be (100-10)% = (90% = 0.9) of the amount of cheetah's the previous year. Since it is going on for 11 years it's exponential. 

1750 * (0.9)^11 

= 549.17

Rounding to the nearest integer we get 549 cheetahs. 

Answer: 

About 549 cheetahs

Thanks!

We have  the points (-1,-1)  and (1,0)  on this graph if we  had a solid line

The slope  of  this line  is  [ 0 - - 1 ] / [ 1  - -1] =  [1]/ [2]  =  1/2

So....using  the point (1,0)  and  the slope  we have either  that

y < (1/2) ( x - 1)     or    y  > (1/2)( x - 1)

Since  the point (0,0)  is in the  feasible region, note  that   0 > (1/2) ( 0 - 1)  =  0 > -1/2  is true

So....the inequality  is

y > (1/2) ( x - 1)

y >  (1/2)x  - 1/2

(1/2)x - y - 1/2 <  0            multiply through by 2

x - 2y - 1  <  0

That is a great observation! If that is the pattern, then you should be able to solve the problem from there!

The first quadratic does not factor so we can skip that one for now...

The second quadratic factors into 3(x+7)(x−8) which doesn't have any repeating roots so this is not the answer.

The third quadratic does not factor so we can skip that one for now...

The fourth quadratic factors into (5x−3)(5x−3) which has repeating roots so this is our answer.

The fifth quadratic factors into (x−2)(x−12) which doesn't have any repeating roots so this is not the answer.

Hope it helps!

I mean the numbers are in order of 3 C 3, 4 C 3, 5 C 3...

To find the arc length you have to use the following formula: 

2*pi*r*(arcmeasure)/360

Let me explain this formula a bit...

2*pi*r, is the circumference of the circle, and we are finding a sector of that. 

The sector is the fraction (arcmeasure/360), which is how much it takes up of the circle.

In this case, the arcmeasure is 180, since it's a straight angle, so the following equation would be 

2*pi*r*(180/360)

= 2*1/2*pi*r

= pi*r

The radius isn't given, so if it was the arc length of the semicircle would be pi times the radius. 

Enjoy!! :) :)

To find the arc length you have to use the following formula: 

2*pi*r*(arcmeasure)/360

Let me explain this formula a bit...

2*pi*r, is the circumference of the circle, and we are finding a sector of that. 

The sector is the fraction (arcmeasure/360), which is how much it takes up of the circle.

In this case, the arcmeasure is 180, since it's a straight angle, so the following equation would be 

2*pi*r*(180/360)

= 2*1/2*pi*r

= pi*r

The radius isn't given, so if it was the arc length of the semicircle would be pi times the radius. 

Enjoy!! :) :)

I'm not seeeing any pattern here, so we should probably look for something else...

For this problem, we wouldn't need the y-intercept, so we could go straight from the information to construct of final answer instead of having to make an equation in slope-intercept form.

Is there 20 arrangements if a is 10? 

Since the brick path is around the fountain, which is circular and has a diameter of 7 (radius: 3.5), we can find the circumference of the circle.

Circumference: pi * 2 * r

= pi * 2 * 3.5

= 7 pi

= 7 * 3.14

= 21.98

Answer: 

21.98 ft

I'm going to answer the solution in Point-Slope and Slope-Intercept. 

Slope - Intercept form is this: y = mx + b

We know that (x, y) is a point on the line, m is the slope, and b is the y-intercept. 

From the given information we have that it has a slope of 2 (m = 2), and one of the points on the line is (3, 9) which means that (x = 3, y = 9)

Plugging in we get:

y = mx + b

9 = 2*3 + b

9 = 6 + b

b = 3

Therefore the y-intercept is 3. After having this information we can construct our equation.

We now have to put this in the point-slope form which is: (y - y1) = m(x - x1)

Transferring what we had before we can plugin and get y - 9 = 2(x - 3)

Comment down if you have any queries or any questions on the solution. Tell me as soon as possible. Good luck in your Algebra Pursuit!! :)

Enjoy!! I put the answer in both point-slope and slope-intercept if you haven't noticed!! :)

So our answer is:

y - 9 = 2(x - 3)​

Look at  the  graph here...https://www.desmos.com/calculator/nofaug4qzu

And note  that  

f(1)  =  -1       f(4)  = -2     f(7)  = -2     f(10)  = -2

f(2)  = -1        f(5) = -2      f(8) = -2      f(11)  =  -2

f(3)  = -1        f(6) = -2      f(9)  = -2     f(12)  = -2

f(13)  = -3

...

f(1000) = -3

This  implies that  the sum is   (-1)*3 + (-2)*9  + (1000 - 13 + 1) (-3)   =  -2985

To solve this question, you have to make systems of equations, for each point. 

As you probably know, the equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (x, y) is a point on the circle, and (h, k) the center of the circle, and r is the radius. 

Now what you have to do is to plug in the values of x and y, and set them equal to each other since we know that the radius is constant. 

First of all, you can easily find out the diameter and the radius of the circle, because you have two points where the y-coordinates are constant, so then finding that you can do |-2 -2| which gives you 4, so now you know that the diameter is 4, and the radius is 2. 

From this plugging in (-2, 0) you get, 

(-2 - h)^2 + (0 - k)^2 = r^2

(-2 - h)^2 + (0 - k)^2 = 4

I'm sure that after you have the idea from here you can distribute, and make 2 more similar equations, then equate and solve. Comment down, if you need another hint or another part of the solution!! 

Good Luck with your Geometry Passion!! :)

:) 

Given the point (3, 9) and the slope of 2, we get y-9= 2(x-3) which is your answer in point-slope form.

Find a linear inequality with the following solution set. Each grid line represents one unit.

https://latex.artofproblemsolving.com/4/8/2/48234b947b623e95b3f920ee2e9ce6cb24dcfa4b.png

Give your answer in the form $ax+by+c>0$ or $ax+by+c\geq0$ where  a and b and c are integers with no common factor greater than 1.)

There are 10 arrangements if a is 11...