Solving the Polynomial Equation (x^4 - 5x^2 - 4 0) and Finding Its Roots

When faced with a polynomial equation like (x^4 - 5x^2 - 4 0), one of the effective strategies is to simplify the problem by introducing a substitution. This guide will walk you through the process of solving this polynomial equation using algebraic methods and the quadratic formula.

Introduction to the Polynomial Equation

The given polynomial equation is:

[x^4 - 5x^2 - 4 0]

While the equation appears complex at first glance, we can simplify it through a substitution. Let’s set (y x^2), which transforms the equation into a more manageable form:

[y^2 - 5y - 4 0]

Solving the Quadratic Equation Using the Quadratic Formula

To solve the quadratic equation (y^2 - 5y - 4 0), we can use the quadratic formula:

[y frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Here, (a 1), (b -5), and (c -4).

Calculating the Discriminant

First, calculate the discriminant (D):

[D b^2 - 4ac (-5)^2 - 4 cdot 1 cdot (-4) 25 16 41]

Solving for (y)

Now, substitute the values into the quadratic formula:

[y frac{-(-5) pm sqrt{41}}{2 cdot 1} frac{5 pm sqrt{41}}{2}]

Thus, the solutions for (y) are:

Reverting Back to (x)

Since (y x^2), we can now revert back to (x) by solving (x^2 y).

For (y frac{5 sqrt{41}}{2}):

[x^2 frac{5 sqrt{41}}{2}]

Thus, the solutions are:

For (y frac{5 - sqrt{41}}{2}):

[x^2 frac{5 - sqrt{41}}{2}]

Thus, the solutions are:

Final Solutions

The complete set of solutions for the polynomial equation (x^4 - 5x^2 - 4 0) is:

Conclusion

By transforming the given polynomial equation into a quadratic form using the substitution (y x^2), we were able to solve for the roots effectively. The roots of the polynomial equation (x^4 - 5x^2 - 4 0) are given by the expressions above.

Additional Keywords

Keywords: polynomial equation, quadratic formula, roots of a polynomial