Solving the Equation x4 - 26x2 25 0: A Step-by-Step Guide
Understanding and solving complex polynomial equations can be a challenging task. This article provides a detailed and clear step-by-step process to solve the equation x4 - 26x2 25 0. We will employ a practical method using the substitution technique to transform the given equation into a more manageable quadratic form and then solve for the values of x.
Introduction to the Equation
The given equation is a bi-quadratic equation, which is a polynomial equation involving terms with an even power of x. Such equations can often be simplified by substituting x2 y, making the equation quadratic in terms of y.
Step-by-Step Solution
Step 1: Substitute x2 y in the given equation. This transforms the equation into a quadratic form.
x4 - 26x2 25 0 becomes:
y2 - 26y 25 0
This is now a standard quadratic equation in the variable y.
Step 2: Apply the quadratic formula to solve for y in the equation y2 - 26y 25 0. The quadratic formula is given by:
y frac{-b pm sqrt{b^2 - 4ac}}{2a}
Here, a 1, b -26, and c 25. Plugging these values into the quadratic formula:
y frac{26 pm sqrt{(-26)^2 - 4 cdot 1 cdot 25}}{2 cdot 1}
Simplifying under the square root:
y frac{26 pm sqrt{676 - 100}}{2}
y frac{26 pm sqrt{576}}{2}
y frac{26 pm 24}{2}
Step 3: Solve for the values of y based on the above formula:
y_1 frac{26 24}{2} frac{50}{2} 25 y_2 frac{26 - 24}{2} frac{2}{2} 1Step 4: Substitute back the values of y to find the corresponding values of x. Recall that y x^2 in the original substitution.
For y 25, x^2 25 implies x pm 5. For y 1, x^2 1 implies x pm 1.The complete solution set for the equation x^4 - 26x^2 25 0 is thus: S {-5, -1, 1, 5}
Conclusion
Solving bi-quadratic equations can be simplified by recognizing a substitution that transforms the equation into a quadratic form. This process involves a clear and methodical approach, which can be applied to similar complex polynomial equations. Understanding the technique of solving quadratic equations and its application in transforming bi-quadratic equations makes these problems more tractable.