Solve equations using a quadratic pattern



Solve.
x4–4x2+4=0
Write your answer as a list of values separated by commas. Simplify all radicals. 



The given equation is:

\[
x^4 - 4x^2 + 4 = 0
\]

This equation can be solved using a substitution. Let \( y = x^2 \). Then the equation becomes:

\[
y^2 - 4y + 4 = 0
\]

Now, solve this quadratic equation:

\[
y^2 - 4y + 4 = (y - 2)^2 = 0
\]

This gives \( y = 2 \).

Since \( y = x^2 \), we substitute back:

\[
x^2 = 2
\]

Taking the square root of both sides:

\[
x = \pm \sqrt{2}
\]

Therefore, the solutions are:

\[
x = \sqrt{2}, x = -\sqrt{2}
\]

So, the answer is:

\[
\boxed{\sqrt{2}, -\sqrt{2}}
\]

Would you like further details or clarifications?

Here are five related questions for further exploration:
1. How would the solution change if the constant term was different?
2. Can all quartic equations be solved by substitution?
3. What happens if you change the equation to \(x^4 - 4x^2 - 4 = 0\)?
4. How can you verify solutions for quadratic and quartic equations?
5. How does solving quartic equations differ from solving cubic equations?

**Tip:** Substitution is a helpful technique when dealing with higher-degree polynomials with quadratic-like structures.

Solve equations using a quadratic pattern. Solve. x^4 – 4x^2 + 4 = 0. Write your answer as a list of values separated by commas. Simplify all radicals.

Learn how to solve the quartic equation x^4 - 4x^2 + 4 = 0 using the substitution method. This detailed solution walks through the steps of solving quartic equations by transforming them into quadratic form, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation