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` I wonder how could I solve a biquadratic equations of general structure "ax^4+bx^3+cx^2+dx+e" . Can any one of you pls help me.`

7 years ago

Solving a quartic equation

Special casesConsider the quartic

If

then

,

so zero is a root. To find the other roots, we can divide by and solve the resultin cubic equation

If

then

,

so 1 is a root. Similarly, if

that is,

then -1 is a root.

When 1 is a root, we can divide

by

and get

where is a cubic polynomial, which may be solved to find 's other roots. Similarly, if -1 is a root,

where is some cubic polynomial.

If

then −*k* is a root and we can factor out ,

And if

then both and are roots Now we can factor out and get

To get *Q* 's other roots, we simply solve the quadratic factor.

**Biquadratic equations**

If then

We call such a polynomial a **biquadratic**, which is easy to solve.

Let Then *Q* becomes a quadratic *q* in z,

Let and be the roots of *q*. Then the roots of our quartic *Q* are

**Quasi-symmetric equations**

Steps:

1) Divide by *x*^{ 2}.

2) Use variable change *z* = *x* + *m*/*x*.

To begin, the quartic must first be converted to a *depressed quartic*.

Converting to a depressed quartic

Let

be the general quartic equation which it is desired to solve. Divide both sides by *A*,

The first step should be to eliminate the *x*^{3} term. To do this, change variables from *x* to *u*, such that

- .

Then

Expanding the powers of the binomials produces

Collecting the same powers of *u* yields

Now rename the coefficients of *u*. Let

The resulting equation is

which is a **depressed quartic equation**.

If then we have a biquadratic Equation, which (as explained above) is easily solved; using reverse substitution we can find our values for *x*.

If then one of the roots is and the other roots can be found by dividing by *u*, and solving the resulting equation,

Using reverse substitution we can find our values for *x*.

Regards,

Rajat,

Askiitians Expert

7 years ago

Dear Askiitians Expert,Many thanks for this method, I found very interesting. I belive that it could be generelized to higher degree equations too.

7 months ago

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