Solving Biquadratic Equations: A Step-by-Step Guide

Biquadratic equations, often seen in advanced algebra and calculus, can appear daunting at first glance. However, they can be simplified into a more manageable form using a clever substitution. In this guide, we'll walk through the process of solving the biquadratic equation 5x^4 5x^1 - 7 0 using the substitution method. This approach will make the equation easier to handle and solve.

Understanding Biquadratic Equations

Biquadratic equations are polynomial equations of the fourth degree. A standard biquadratic equation has the form a x^4 b x^2 c 0. However, in our example, the equation is 5x^4 5x - 7 0, which at first may seem non-standard. By making a substitution, we can transform it into a quadratic form, making it easier to solve.

The Substitution Method

The key to solving biquadratic equations lies in the smart use of substitution. Let's define a new variable t 5x^1. This substitution simplifies the equation into a more familiar form.

Step 1: Substitution

Begin by substituting t 5x^1 into the original equation:

5 x 4 5 x - 7 0

After substituting 5x^1 t, we get:

5 t 2 - 7 0

This simplifies to:

t t - 7 0

or equivalently:

t t 7

Step 2: Solving the Quadratic Equation

Now, we need to solve the quadratic equation t^2 - 7 0 for t. t t - 7 0

This can be factored as:

t - 7 t 7 0

So the solutions for t are:

t -7 or t 1

Step 3: Back Substitution

Now that we have the values for t, we need to recall that t 5x^1. So we substitute back to find x.

t 1 gives 5x^1 1, so x 0. t -7 gives 5x^1 -7, so x -7/5.

Therefore, the solutions for the original biquadratic equation are x 0 and x -7/5.

Summary

Solving biquadratic equations involves a clever substitution to transform the problem into a quadratic form. By defining a new variable, such as t 5x^1, we can simplify the equation and solve it more effectively.

Practice and Resources

To further your understanding and practice solving biquadratic equations, consider the following resources:

Online tutorials and video lessons on biquadratic equations. Interactive problem sets and solutions available on math websites and educational platforms. Practice problems in textbooks and math workbooks.

Remember, the key to mastering biquadratic equations is consistent practice and understanding the substitution method. Happy solving!