Question

how to solve biquadratic equation.

Answer 1

A quadratic equation is an equation where the highest exponent of a variable is 2. There are three main ways to solve quadratic equations: 1) to factor the quadratic equation if you can do so, 2) to use the quadratic formula, or 3) to complete the square.

Answer 2

We can solve Any Quartic Equation of the form

$\sf f(x) = a {x}^{4} + b {x}^{3} + c {x}^{2} + dx + e$

We simply need to solve the resolvent cubic generated by the Quartic equation by completing the square twice.

$\sf8 {a}^{3} {(z)}^{3} - 4 {a}^{2} c {(z)}^{2} + (2abd - 8 {a}^{2} e)(z) + (4ace - a {d}^{2} - {b}^{2} e) = 0$

Therefore, the Four roots are given by the roots of the Two Quadratic equations:

$\sf{x}^{2} + ( \frac{b + \sqrt{ {b}^{2} - 4ac + 8 {a}^{2} z }}{2a} )x + (z + \sqrt{ {z}^{2} - \frac{e}{a} } ) = 0$

$\sf {x}^{2} + ( \frac{b - \sqrt{ {b}^{2} - 4ac + 8 {a}^{2} z}}{2a} )x + (z - \sqrt{ {z}^{2} - \frac{e}{a} } ) = 0$

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