X to the power of 4 minus 3x to the power of 2 plus 1 equal to 0

To solve the equation \( x^4 - 3x^2 + 1 = 0 \), we can use a substitution method to transform it into a quadratic equation.

1. Let \( y = x^2 \). This means \( y^2 = x^4 \).
2. Substitute \( y \) into the equation:
   \[
   y^2 - 3y + 1 = 0
   \]

Now we need to solve the quadratic equation \( y^2 - 3y + 1 = 0 \).

Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -3 \), and \( c = 1 \):

\[
y = \frac{3 \pm \sqrt{9 - 4}}{2}
\]
\[
y = \frac{3 \pm \sqrt{5}}{2}
\]

So, we have two solutions for \( y \):
\[
y_1 = \frac{3 + \sqrt{5}}{2}
\]
\[
y_2 = \frac{3 - \sqrt{5}}{2}
\]

Recall that \( y = x^2 \). Thus, we need to solve for \( x \) by taking the square roots of \( y_1 \) and \( y_2 \):

1. \( x^2 = \frac{3 + \sqrt{5}}{2} \)
   \[
   x = \pm \sqrt{\frac{3 + \sqrt{5}}{2}}
   \]

2. \( x^2 = \frac{3 - \sqrt{5}}{2} \)
   \[
   x = \pm \sqrt{\frac{3 - \sqrt{5}}{2}}
   \]

So, the solutions to the original equation \( x^4 - 3x^2 + 1 = 0 \) are:
\[
x = \pm \sqrt{\frac{3 + \sqrt{5}}{2}}, \quad x = \pm \sqrt{\frac{3 - \sqrt{5}}{2}}
\]

Would you like any more details on this solution or have any questions?

Here are 8 relative questions to expand your understanding:

1. How do you derive the quadratic formula?
2. What are the steps to simplify radicals in solutions?
3. How do you check if the roots found are correct?
4. Can the substitution method be used for other polynomial equations?
5. What is the significance of imaginary roots?
6. How do you graph the function \( y = x^4 - 3x^2 + 1 \)?
7. What other methods can be used to solve quartic equations?
8. How can this problem be solved using numerical methods?

**Tip:** When solving higher-degree polynomials, look for substitutions or factorizations that simplify the equation into more manageable forms.

Learn how to solve the polynomial equation x^4 - 3x^2 + 1 = 0 using the substitution method. This detailed solution includes key algebraic concepts such as polynomial equations and the quadratic formula. Suitable for high school students.

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