Solve The Equation:${ 8x^4 - 18x^2 = 0 }$
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Introduction
Solving equations is a fundamental concept in mathematics, and it's essential to understand various techniques to tackle different types of equations. In this article, we will focus on solving the equation 8x^4 - 18x^2 = 0, which is a quartic equation. We will break down the solution step by step and provide a clear explanation of each step.
Understanding the Equation
The given equation is 8x^4 - 18x^2 = 0. This equation can be rewritten as 8x^4 = 18x^2. We can see that the equation has a common factor of x^2 on both sides. We can factor out x^2 from both sides to get 8x2(x2 - 2.25) = 0.
Factoring the Equation
We can factor the equation further by recognizing that x^2 - 2.25 is a difference of squares. We can rewrite it as (x - √2.25)(x + √2.25) = 0.
Solving for x
Now that we have factored the equation, we can solve for x. We can set each factor equal to zero and solve for x.
- 8x^2 = 0
- x^2 - 2.25 = 0
Solving the first equation, we get x = 0.
Solving the second equation, we get x^2 = 2.25, which gives us x = ±√2.25.
Simplifying the Solutions
We can simplify the solutions by evaluating the square root of 2.25. √2.25 = √(2.5^2) = 2.5.
So, the solutions to the equation are x = 0, x = 2.5, and x = -2.5.
Conclusion
In this article, we solved the equation 8x^4 - 18x^2 = 0 by factoring and then solving for x. We broke down the solution step by step and provided a clear explanation of each step. We also simplified the solutions by evaluating the square root of 2.25.
Frequently Asked Questions
Q: What is the difference between a quadratic equation and a quartic equation?
A: A quadratic equation is a polynomial equation of degree two, while a quartic equation is a polynomial equation of degree four.
Q: How do I factor a quartic equation?
A: To factor a quartic equation, you can look for common factors, recognize patterns such as the difference of squares, and use techniques such as grouping.
Q: What is the significance of the solutions to the equation?
A: The solutions to the equation represent the values of x that satisfy the equation. In this case, the solutions are x = 0, x = 2.5, and x = -2.5.
Additional Resources
- Khan Academy: Solving Quadratic Equations
- Mathway: Solving Quartic Equations
- Wolfram Alpha: Solving Equations
Final Thoughts
Solving equations is a fundamental concept in mathematics, and it's essential to understand various techniques to tackle different types of equations. In this article, we solved the equation 8x^4 - 18x^2 = 0 by factoring and then solving for x. We broke down the solution step by step and provided a clear explanation of each step. We also simplified the solutions by evaluating the square root of 2.25.
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Introduction
In our previous article, we solved the equation 8x^4 - 18x^2 = 0 by factoring and then solving for x. We broke down the solution step by step and provided a clear explanation of each step. We also simplified the solutions by evaluating the square root of 2.25. In this article, we will provide a Q&A section to address some common questions and concerns related to solving the equation.
Q&A
Q: What is the difference between a quadratic equation and a quartic equation?
A: A quadratic equation is a polynomial equation of degree two, while a quartic equation is a polynomial equation of degree four.
Q: How do I factor a quartic equation?
A: To factor a quartic equation, you can look for common factors, recognize patterns such as the difference of squares, and use techniques such as grouping.
Q: What is the significance of the solutions to the equation?
A: The solutions to the equation represent the values of x that satisfy the equation. In this case, the solutions are x = 0, x = 2.5, and x = -2.5.
Q: Can I use the quadratic formula to solve a quartic equation?
A: No, the quadratic formula is used to solve quadratic equations, not quartic equations. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, and it is only applicable to quadratic equations.
Q: How do I determine the number of solutions to a quartic equation?
A: To determine the number of solutions to a quartic equation, you can use the discriminant. The discriminant is the expression under the square root in the quadratic formula, and it can be used to determine the number of real solutions to a quadratic equation. For a quartic equation, you can use the discriminant of the quadratic factors to determine the number of real solutions.
Q: Can I use a calculator to solve a quartic equation?
A: Yes, you can use a calculator to solve a quartic equation. Many calculators have built-in functions to solve polynomial equations, including quartic equations.
Q: What is the difference between a rational root and an irrational root?
A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction.
Q: How do I determine if a root is rational or irrational?
A: To determine if a root is rational or irrational, you can use the rational root theorem. The rational root theorem states that if a rational root exists, it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Additional Resources
- Khan Academy: Solving Quadratic Equations
- Mathway: Solving Quartic Equations
- Wolfram Alpha: Solving Equations
Final Thoughts
Solving equations is a fundamental concept in mathematics, and it's essential to understand various techniques to tackle different types of equations. In this article, we provided a Q&A section to address some common questions and concerns related to solving the equation 8x^4 - 18x^2 = 0. We hope that this article has been helpful in clarifying any doubts and providing a better understanding of solving quartic equations.
Common Mistakes to Avoid
- Not factoring the equation properly
- Not using the correct techniques to solve the equation
- Not checking for rational or irrational roots
- Not using a calculator or computer algebra system to check the solutions
Conclusion
In this article, we provided a Q&A section to address some common questions and concerns related to solving the equation 8x^4 - 18x^2 = 0. We hope that this article has been helpful in clarifying any doubts and providing a better understanding of solving quartic equations. Remember to always follow the correct techniques and to check your solutions carefully to avoid common mistakes.