Alan

You will need a numerical method to find the root. One possibility is the Newton-Raphson method:

y(x) = x^3 + 3x^2 + 2x + 1

dy/dx(x) = 3x^2 + 6x + 2

Start with an initial guess of x₀ (say, -2.5)

Then use the following iterative process:  x_n+1 = x_n - y(x_n)/dy/dx(x_n) repeatedly until you get a sufficiently accurate result.

The polynomial  x^3 + 3x^2 + 2x + 1 has just one real root, and doesn't factorise nicely - see the graph below:

Have a look at this Youtube video:  https://www.youtube.com/watch?v=bYZWW0fSLJU 

Small correction:  in my fourth expression above the term beginning 1/8 should be preceded by a minus sign, not a plus sign.

Pair them up as follows:   (53 + 47) + (18 + 82) → 100 + 100 → 200

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Oops, sorry, I should have put in more explanation:

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Hope this makes it clearer!

Here's an alternative approach:

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cos(x) = 2cos^2(x/2) - 1

Hence you have    2cos^2(x/2) - 1 = cos(x/2)

                              2cos^2(x/2) - cos(x/2) - 1 = 0

                              (2cos(x/2) + 1)(cos(x/2) -1) = 0

Possible solutions are:    cos(x/2) = -1/2    and    cos(x/2) = 1

Can you take it from here?

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d = number of dragons, r = number of riders

Heads:    d + r = 44     (1)

Legs:      4d + 2r = 128   (2)

From (1) we have r = 44 - d     (3)

Use (3) in (2):   4d + 2(44 - d) = 128

                         2d + 88 = 128

                         2d = 40

                           d = 20

Use this in (3):   r = 44 - 20 so r = 24

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