# A strange equation V

Algebra Level 5

The seven real roots of the equation $$x^7+28x^6+308x^5+1680x^4+4704x^3+6272x^2+3136x-128\sqrt{2}+256 = 0$$ are, in ascending order: $x_{1}=-a-b\cos\left(\dfrac{c\pi}{d}\right)$ $x_{2}=-a-b\cos\left(\dfrac{e\pi}{d}\right)$ $x_{3}=-a-b\cos\left(\dfrac{f\pi}{d}\right)$ $x_{4}=-a-b\cos\left(\dfrac{g\pi}{d}\right)$ $x_{5}=-a+b\cos\left(\dfrac{h\pi}{d}\right)$ $x_{6}=-a+j\sqrt{j}$ $x_{7}=-a+b\cos\left(\dfrac{k\pi}{d}\right)$ All the variables are positive integer numbers, and the angles are in radians between $$0$$ and $$\frac{\pi}{2}$$. Find $$a+b+c+d+e+f+g+h+j+k$$.

You may also try Part VI.

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