Choosing the Right Substitution

Oct 13, 2025 by magicalceylontours.com 44 views

Introduction

Quadratic equations are a fundamental concept in algebra, and they can be used to solve a wide range of problems in mathematics and other fields. However, not all equations are in the standard quadratic form, and sometimes we need to use substitution to rewrite them in a more manageable form. In this article, we will explore how to rewrite the equation $$x^8 - 3x^4 + 2 = 0$$ as a quadratic equation using substitution.

Understanding the Problem

The given equation is a quartic equation, which means it has a degree of 4. To rewrite it as a quadratic equation, we need to find a substitution that will reduce the degree of the equation to 2. This can be done by introducing a new variable, say u, and expressing x in terms of u.

Choosing the Right Substitution

To choose the right substitution, we need to analyze the given equation and identify the term that has the highest power of x. In this case, the term with the highest power of x is $$x^8$$. We need to find a substitution that will eliminate this term and reduce the degree of the equation.

Option A: $u = x^2$

Let's consider the first option, which is $u = x^2$ . If we substitute $u = x^2$ into the given equation, we get:

u^4 - 3u^2 + 2 = 0

This is still a quartic equation, and we have not reduced the degree of the equation. Therefore, this option is not correct.

Option B: $u = x^4$

Now, let's consider the second option, which is $u = x^4$ . If we substitute $u = x^4$ into the given equation, we get:

u^2 - 3u + 2 = 0

This is a quadratic equation, and we have successfully reduced the degree of the equation. Therefore, this option is correct.

Option C: $u = x^8$

Let's consider the third option, which is $u = x^8$ . If we substitute $u = x^8$ into the given equation, we get:

u - 3u^{1/2} + 2 = 0

This is not a quadratic equation, and we have not reduced the degree of the equation. Therefore, this option is not correct.

Option D: $u = x^{16}$

Finally, let's consider the fourth option, which is $u = x^{16}$ . If we substitute $u = x^{16}$ into the given equation, we get:

u^{1/4} - 3u^{1/8} + 2 = 0

This is not a quadratic equation, and we have not reduced the degree of the equation. Therefore, this option is not correct.

Conclusion

In conclusion, the correct substitution to rewrite $$x^8 - 3x^4 + 2 = 0$$ as a quadratic equation is $u = x^4$ . This substitution reduces the degree of the equation from 4 to 2, making it a quadratic equation.

Final Answer

The final answer is B. $u = x^4$ .

Introduction

In our previous article, we discussed how to rewrite the equation $$x^8 - 3x^4 + 2 = 0$$ as a quadratic equation using substitution. We also provided the correct substitution, which is $u = x^4$ . In this article, we will answer some frequently asked questions (FAQs) about rewriting this equation as a quadratic equation.

Q: What is the purpose of rewriting the equation as a quadratic equation?

A: The purpose of rewriting the equation as a quadratic equation is to make it easier to solve. Quadratic equations have a well-known formula for solving them, which is the quadratic formula. By rewriting the equation as a quadratic equation, we can use this formula to find the solutions.

Q: Why is the substitution $u = x^4$ the correct one?

A: The substitution $u = x^4$ is the correct one because it reduces the degree of the equation from 4 to 2, making it a quadratic equation. This substitution eliminates the term with the highest power of x, which is $$x^8$$.

Q: Can I use other substitutions to rewrite the equation as a quadratic equation?

A: Yes, you can use other substitutions to rewrite the equation as a quadratic equation. However, the substitution $u = x^4$ is the most straightforward and easiest to use.

Q: How do I know if a substitution is correct or not?

A: To determine if a substitution is correct or not, you need to check if it reduces the degree of the equation. If the substitution reduces the degree of the equation, then it is correct.

Q: Can I use the substitution $u = x^2$ to rewrite the equation as a quadratic equation?

A: No, you cannot use the substitution $u = x^2$ to rewrite the equation as a quadratic equation. This substitution does not reduce the degree of the equation, and it will result in a quartic equation.

Q: Can I use the substitution $u = x^8$ to rewrite the equation as a quadratic equation?

A: No, you cannot use the substitution $u = x^8$ to rewrite the equation as a quadratic equation. This substitution does not reduce the degree of the equation, and it will result in a non-quadratic equation.

Q: Can I use the substitution $u = x^{16}$ to rewrite the equation as a quadratic equation?

A: No, you cannot use the substitution $u = x^{16}$ to rewrite the equation as a quadratic equation. This substitution does not reduce the degree of the equation, and it will result in a non-quadratic equation.

Q: How do I apply the substitution $u = x^4$ to rewrite the equation as a quadratic equation?

A: To apply the substitution $u = x^4$ , you need to replace $x^4$ with $u$ in the original equation. This will result in a quadratic equation in terms of $u$ .

Q: What is the resulting quadratic equation after applying the substitution $u = x^4$ ?

A: The resulting quadratic equation after applying the substitution $u = x^4$ is $u^2 - 3u + 2 = 0$ .

Conclusion

In conclusion, rewriting the equation $$x^8 - 3x^4 + 2 = 0$$ as a quadratic equation using substitution is a useful technique for solving equations. By using the correct substitution, $u = x^4$ , we can reduce the degree of the equation from 4 to 2, making it a quadratic equation. We hope that this article has answered some of the frequently asked questions about rewriting this equation as a quadratic equation.

Final Answer

The final answer is B. $u = x^4$ .