Answers

2015-06-07T23:08:40+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
Cubic equation / cubic polynomial  :  x³ - 3 p x - 2 q = 0
Solution is given by:     
x_k,\ \ k=0,1,2\\\\x_k=2\ *\ \sqrt{p}\ *\ Cos[\frac{1}{3}\ *\ Cos^{-1}(\frac{q}{p^{\frac{3}{2}}})-\frac{2\pi k}{3}]

Or, by:
x = [q+\sqrt{q^2-p^2}]^{\frac{1}{3}}+[q-\sqrt{q^2-p^3}]^{\frac{1}{3}}

Given  y³ - 7 y + 6 = 0
So substituting      p = 7/3    and    q = -3  we get :

x_0=2*\sqrt{7/3}\ Cos[\frac{1}{3}*Cos^{-1}(\frac{-3}{(7/3)^{\frac{3}{2}}})-\frac{2\pi *0}{3}]=2\\\\x_1=2*\sqrt{7/3}\ Cos[\frac{1}{3}*Cos^{-1}(\frac{-3}{(7/3)^{\frac{3}{2}}})-\frac{2\pi *1}{3}]=1\\\\x_2=2*\sqrt{7/3}\ Cos[\frac{1}{3}*Cos^{-1}(\frac{-3}{(7/3)^{\frac{3}{2}}})-\frac{2\pi *1}{3}]=-3

We can try to find the solutions of  the polynomial equation: y³ - 7 y + 6 = 0  by
  constant term / coefficient of y³:    +or -  6 /1  or its factors :  + or - of 6, 3, 2, 1
So we can try these 8 possibilities.  It so happens that 1, 2, -3 are the roots.

By using the other formula :

x = [ -3 + \sqrt{(-3)^2-(7/3)^3} ]^{\frac{1}{3}} + [q - \sqrt{(-3)^2-(7/3)^{\frac{1}{3}}}]\\\\=[-3+1.9245\ i]^{\frac{1}{3}}+[-3-1.9245\ i]^{\frac{1}{3}}

This solution gives complex numbers.  So it requires to convert them in the polar coordinate form  e^{i\theta}  and then take cube root and then add the two cube roots.
Then we get the same answers.

1 5 1
click on thanks button above please