proyaop

Instead of doing the entire problem by getting each root exactly, why not try and use Vieta's formula to get ab and a + b instead? 

ab = 18 and a + b = -5. Thus, a^2 + b^2 = (a + b)^2 - 2ab. Then a^2 + b^2 = 25 - 36.

Thus, a^2 + b^2 = -11. 

Instead of listing, why not write an equation? If we multiply all 4 equations together, we get:

$a^2b^2c^2d^2=36^2$

Thus abcd = 36. 

If you need any help, just think about trying to substitute variables in terms of other variables such as a = 6/c, and try doing that and plugging it back into the main equation abcd = 36. 

Hopefully this method (the normal method) will help you find ALL sets faster.

The best way is to use this method in the future, or else for more difficult problems it may be almost impossible to brainstorm every single set. Good luck...

:D

You will end up with more complicated polynomials, and the person who asked the question may not be able to fully expand it out or solve it with a cubic equation.

Though if you do factor it out it may be a better strategy if you know how... :)

So a^3 + b^3 = (a + b)(a^2 - ab + b^2). 

Substituting in, we have 812 = 14(a^2 - ab + b^2).

Simplifying we have 58 = a^2 - ab + b^2.

a^2 - ab + b^2 = (a + b)(a + b) - 3ab. 

Substituting in, we have 58 = (14)(14) - 3ab.

Now we have ab = 46.

Then we also have a^2 - ab + b^2 = (a - b)(a - b) + ab.

Thus, 58 = (a - b)^2 + 46.

(a - b)^2 = 12.

$a - b = \pm{2\sqrt{3}}$

Now we can sove for a. If we add the two equations together, we get 2a = 14 + 2sqrt(3)

Thus, $a = 7 + \sqrt{3}$, and $a = 7 - \sqrt{3}$.

Substituting in, we have $b = 7 - \sqrt{3}$, and $b = 7 - 3\sqrt{3}$.

Thus, our possible for solutions $(a, b)$are:

$(7 + \sqrt{3}, 7 - \sqrt{3})$

$(7 - \sqrt{3}, 7 - 3\sqrt{3})$

First to calculate probability, we have total number of restricted ways/total number of possibilities. Since he moves 8 steps, and there are only 2 possible locations adjacent to which he can go, there are 8^2 = 64 TOTAL possibilities. 

There are 8 steps, and 5 other vertices in the hexagon to go from.

Maybe this information could get you started... :) 

I noticed I see five '5^b's. Thus, the equation would be:

$5^{b+1}=25^{b-1}$

25 = 5^2, so the equation would be:

$5^{b+1}=5^{2b-2}$

Since they are equal and they have the same base, then:

b + 1 = 2b - 2.

Solving for b, we get $b=3$.

I got the same answer as CPhill. 

Since the radii are equal length, thus the other angle is 50 degrees, and the central angle is 80 degrees. We can use law of cosines, so cos 80 is approximately 0.17, and the radii are 18 units each.

That means the length of the chord^2 = 18^2 + 18^2 - 2*18*18*cos80 degrees.

Length of chord^2 = 648 - 648*0.17.

We get:

Length of chord^2 = 535.476.

Length of chord is 23.14.

Thus, in terms of pi it would be approximately 7.3658pi.

:)

I got it.. angle x is 15 degrees. The perpendicular from D to CB is half of CD. The perpendicular from D to AB and the perpendicular from D to CB form a rectangle. Thus, you can do some 30-60-90 shenanigans, and get that the perpendicular from D to AB (point p) is 2 - sqrt(3) times AP. Thus, with the side ratios I discovered, I got x = 15 degrees. :D :D :D

Do the math if you are a bit confused, i big brained it lol :)