If an equation has the shape:
ax^{2n}
+ 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: 3x^{4}
– 123x^{2}
+ 1200 = 0 .
S o l u t i o n . Exchanging: x^{2}
= z , and solving the
equation:
3z
^{2}
– 123z + 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.
