Ziggy

You too!!  

If a has 3 factors, then a will be the square of a prime number (it's factors are 1, a prime number, and the square of the prime number). The smallest square of a prime number is 2^2 = 4. Therefore, if b is divisible by 4 and has 4 factors, than the least possible value of b is 8.

Since we are told that the polynomial x^2+bx+c has exactly one real root, we know that the discriminant (b^2-4ac) is equal to 0. Thus, plugging in a = 1 and b = c-2 into the equation, we get c^2 - 8c + 4 = 0. I think this will give us c = 4 +- 2sqrt(3), so the product of all possible values of c is [4+2sqrt(3)][4-2sqrt(3)] = 16 - 12 = 4.

Here's my solution:

We know that the points that he labels with the number 25 has a distance of sqrt(25) = 5 from the origin. Thus, using the distance formula and calling the points that satisfy the condition (x, y), we have:

sqrt(x^2+y^2) = 5. Sqaring both sides gives us x^2+y^2 = 25. See if you can solve it from there! 

Hint: We have to find the number of points (x, y) that satisfy the equation x^2+y^2 = 25. We also have to remember that x and y can be negative numbers too, but they all have to be real integers. Listing the possibilities out, we get

(x, y) = (0, 5), (5, 0), (0, -5), (-5, 0), (3, 4), (-3, 4), (3, -4), (-3, -4), (4, 3), (-4, 3), (4, -3), and (-4, -3). How many pairs is that? 

12 times (I did this problem yesterday lol)

Sure, I can try! First, we can call the side length of the square "x". The area of this orginal square would be x^2. Next, if the length of the side of the square was doubled, then one side would have a length of 2*x = 2x. The area of this new sqaure would be (2x)^2 = 4x^2. Therefore, the ratio of the areas of the original sqaure and the area of the new sqaure is x^2 / 4x^2 = 1/4, or 1:4. I hope that helped! 

3.1536 × 10^6

First, we cross multiply for all three equations to get 

a^2 = b, b^2 = 2c, and c^2 = 4a. Thus, we know that a = sqrt(b) from the first equation and b = sqrt(2c) from the second. Next, using the last equation, we find that 

c^2 = 4a ---> c^2 = 4sqrt(b). We already found that b = sqrt(2c), so we can substitute it in to get c^2 = 4*sqrt(sqrt(2c)) = 4*∜2c. Further simplifying it, we get c^8 = 512c ---> c^7 = 512, so c equals the seventh root of 512. 

Lastly, we plug this value of c into the third equation to obtain a = about 1.48599429.

The expression (4k + 8)/4 can be written as 4k/4 + 8/4 = 1k + 2 = k + 2. Therefore, a = 1 and b = 2, so a/b = 1/2.