Equations of higher degrees
Biquadratic equation. Cubic equation.
1. 
Some kinds of the higher degrees equations may be solved using a
quadratic equation. Sometimes one can resolve the lefthand side of
equation to factors, each of them is a polynomial of the degree not
higher than second. Then, equaling each of them to zero and solving all
these quadratic and / or linear equations, we’ll receive all roots of
the original equation.
E x a m p l e . Solve an equation: 3 x ^{ 4 } + 6 x ^{ 3 } – 9 x ^{ 2 } = 0 . S o l u ti o n . Resolve the lefthand side of this equation to factors: x ^{ 2 } ( 3 x ^{ 2 } + 6 x – 9 ) . Solve the equation: x ^{ 2 } = 0; it has two equal roots: x _{ 1 } = x _{ 2 } = 0 . Now we solve the equation: 3 x ^{ 2 } + 6 x – 9 = 0, and receive: x _{ 3 } = 1 and x _{ 4 } = – 3 . Thus, the original equation has four roots: x _{ 1 } = x _{ 2 } = 0 ; x _{ 3 } = 1 ; x _{ 4 } = – 3 . 

2. 
If an equation has the shape:
ax ^{ 2 n } + bx ^{ n } + c = 0 ,
it is reduced to an quadratic equation by the exchange:
x ^{ n } = z ;
really, after this exchange we receive: az ^{ 2 } + bz + c = 0 .
E x a m p l e . Consider the equation:
x ^{ 4 } – 13 x ^{ 2 } + 36 = 0 .
Exchange: x ^{ 2 } = z . After this we receive:
z ^{ 2 } – 13 z + 36 = 0 .
Its roots are: z _{ 1 } = 4 and z _{ 2 } = 9. Now we solve the equations: x ^{ 2 } = 4 and x ^{ 2 } = 9 . They have the roots correspondingly: x _{ 1 } = 2 , x _{ 2 } = – 2 , x _{ 3 } = 3 ; x _{ 4 } = – 3 . These numbers are the roots of the original equation ( check this, please ! ).
Any equation of the shape: ax ^{ 4 } + bx ^{ 2 } + c = 0 is called a biquadratic equation. It is reduced to quadratic equations by using the exchange: x ^{ 2 } = z .
E x a m p l e . Solve the biquadratic equation: 3 x ^{ 4 } – 123 x ^{ 2 } + 1200 = 0 .
S o l u t i o n . Exchanging: x ^{ 2 } = z , and solving the equation: 3 z ^{ 2 } – 123 z + 1200 = 0 , we’ll receive:
hence, z _{ 1 } = 25 and z _{ 2 } = 16 . Using our exchange, we receive:
x
^{
2
}
=
25 and
x
^{
2
}
=
16, hence,
x
_{
1
}
=
5,
x
_{
2
}
=
–5,
x
_{
3
}
=
4,
x
_{
4
}
=
– 4.


3. 
A
cubic equation
is the third
degree equation; its general shape is:
The known Cardano’s formulas for solution of this kind equations are very difficult and almost aren’t used in practice. So, we recommend another way to solve the third degree equations.
