Math Problem Statement
x6 − 4x3 − 5 = 0
Solution
The equation is:
Let's solve it step by step.
- We can perform a substitution to simplify the equation. Let , then . This transforms the original equation into:
- Now, we can solve this quadratic equation. The general form of a quadratic equation is , and its solutions are given by:
For , we have:
Now substitute these values into the quadratic formula:
So, the two possible values for are:
- Now, recall that , so we have two equations to solve:
- Solving these gives us the cube roots:
Thus, the real solutions are and .
Do you want more details or have any questions?
Here are five related questions for further practice:
- How would you solve the equation ?
- What are the solutions to ?
- How do you solve ?
- How would you approach solving ?
- What happens when ?
Tip: Substitution is a helpful technique to reduce higher-degree polynomials to simpler forms, making them easier to solve.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Equations
Substitution Method
Formulas
Quadratic equation formula: y = (-b ± √(b^2 - 4ac)) / 2a
Theorems
Quadratic Formula Theorem
Suitable Grade Level
Grades 9-12