Biquadratic Equations

Biquadratic equations are quartic equations with no odd-degree terms:

ax4 + bx2 + c = 0

Solving Biquadratic Equations

To solve biquadratic equations, change x2 = t, x4 = t2; this generates a quadratic equation with the unknown, t:

at2 + bt + c = 0

For every positive value of t there are two values of x, find:

Biquadratic Equation

Biquadratic Equation

Biquadratic Operations

Biquadratic Operations

Biquadratic Operations

Biquadratic Operations

The same procedure can be used to solve the equations of the type:

ax6 + bx3 + c = 0

ax8 + bx4 + c = 0

ax10 + bx5 + c = 0

Biquadratic Equation

Biquadratic Operations

Biquadratic Operations

Biquadratic Equation Solution

Biquadratic Equation Solution

Examples

x4 − 10x2 + 9 = 0

x4 − 10x2 + 9

x2 = t

x4 − 10x2 + 9 = 0

t2 − 10t + 9 = 0

Biquadratic Operations

Biquadratic Equation Solution

Biquadratic Equation Solution


x4 − 61x2 + 900 = 0

Biquadratic Equation

Biquadratic Operations

Biquadratic Operations

Biquadratic Operations

Biquadratic Operations


x4 − 25x2 + 144 = 0

Biquadratic Equation

Biquadratic Equation

Biquadratic Operations

Biquadratic Operations

Biquadratic Operations


x4 − 16x2 − 225 = 0

Biquadratic Equation

Biquadratic Equation

Biquadratic Operations

Biquadratic Operations

Biquadratic Equation Solution