Solve the following equations :
i) `x^(4)-5x^(2)+6=0`

Answer

To solve the equation \( x^4 - 5x^2 + 6 = 0 \), we can start by making a substitution to simplify the equation. Let's denote \( y = x^2 \). This transforms our equation into a quadratic form:

1. **Substitution**:
\[ y^2 - 5y + 6 = 0 \]

2. **Factoring the quadratic**:
We need to factor the quadratic equation. We look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( y \)). The numbers \( -2 \) and \( -3 \) fit this requirement.
\[ (y - 2)(y - 3) = 0 \]

3. **Setting each factor to zero**:
We set each factor equal to zero to find the values of \( y \):
\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

4. **Substituting back for \( x \)**:
Recall that \( y = x^2 \). Therefore, we have:
\[ x^2 = 2 \quad \Rightarrow \quad x = \pm \sqrt{2} \]
\[ x^2 = 3 \quad \Rightarrow \quad x = \pm \sqrt{3} \]

5. **Final solution**:
The complete set of solutions for \( x \) is:
\[ x = \sqrt{2}, \quad x = -\sqrt{2}, \quad x = \sqrt{3}, \quad x = -\sqrt{3} \]

### Summary of Solutions:
The solutions to the equation \( x^4 - 5x^2 + 6 = 0 \) are:
\[ x = \sqrt{2}, \quad x = -\sqrt{2}, \quad x = \sqrt{3}, \quad x = -\sqrt{3} \]