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

Answer

To solve the equation \( x^4 - 5x^2 + 6 = 0 \), we can follow these steps:

### Step 1: Substitute \( x^2 \) with a new variable
Let \( a = x^2 \). Then, the equation can be rewritten as:
\[ a^2 - 5a + 6 = 0 \]

### Step 2: Factor the quadratic equation
Now we need to factor the quadratic equation \( a^2 - 5a + 6 \). We look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( a \)). The numbers \( -2 \) and \( -3 \) work because:
\[ -2 \times -3 = 6 \quad \text{and} \quad -2 + (-3) = -5 \]
Thus, we can factor the equation as:
\[ (a - 2)(a - 3) = 0 \]

### Step 3: Solve for \( a \)
Setting each factor to zero gives us:
\[ a - 2 = 0 \quad \Rightarrow \quad a = 2 \]
\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]

### Step 4: Substitute back to find \( x \)
Recall that we set \( a = x^2 \). Therefore, we have:
1. \( x^2 = 2 \)
2. \( x^2 = 3 \)

### Step 5: Take the square root
Now we take the square root of both sides for each equation:
1. From \( x^2 = 2 \):
\[ x = \pm \sqrt{2} \]
2. From \( x^2 = 3 \):
\[ x = \pm \sqrt{3} \]

### Final Solution
Thus, the complete set of solutions for the original equation \( x^4 - 5x^2 + 6 = 0 \) is:
\[ x = \sqrt{2}, \quad x = -\sqrt{2}, \quad x = \sqrt{3}, \quad x = -\sqrt{3} \]

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