Find the integral values of a for which the equation `x^4-(a^2-5a+6)x^2-(a^2-3a+2)=0` has only real roots

Explanation

To find the integral values of \( a \) for which the equation

\[ x^4 - (a^2 - 5a + 6)x^2 - (a^2 - 3a + 2) = 0 \]

has only real roots, we can follow these steps:

### Step 1: Substitute \( x^2 = t \)

Let \( t = x^2 \). Then the equation becomes:

\[ t^2 - (a^2 - 5a + 6)t - (a^2 - 3a + 2) = 0 \]

### Step 2: Identify the conditions for real roots

For the quadratic in \( t \) to have real roots, the discriminant must be non-negative. The discriminant \( D \) of a quadratic \( at^2 + bt + c = 0 \) is given by:

\[ D = b^2 - 4ac \]

Here, \( a = 1 \), \( b = -(a^2 - 5a + 6) \), and \( c = -(a^2 - 3a + 2) \).

### Step 3: Calculate the discriminant

Calculating the discriminant:

\[ D = (-(a^2 - 5a + 6))^2 - 4 \cdot 1 \cdot (-(a^2 - 3a + 2)) \]

This simplifies to:

\[ D = (a^2 - 5a + 6)^2 + 4(a^2 - 3a + 2) \]

### Step 4: Set the discriminant \( D \geq 0 \)

We need \( D \geq 0 \):

\[ (a^2 - 5a + 6)^2 + 4(a^2 - 3a + 2) \geq 0 \]

### Step 5: Analyze the quadratic in \( t \)

The quadratic in \( t \) will have real roots if:

1. The sum of the roots \( t_1 + t_2 \geq 0 \)
2. The product of the roots \( t_1 t_2 \geq 0 \)

From Vieta's formulas, we know:

- The sum of the roots \( t_1 + t_2 = a^2 - 5a + 6 \)
- The product of the roots \( t_1 t_2 = -(a^2 - 3a + 2) \)

### Step 6: Set conditions for the sum and product of roots

1. For the sum of roots:

\[ a^2 - 5a + 6 \geq 0 \]

Factoring gives:

\[ (a - 2)(a - 3) \geq 0 \]

This implies:

\[ a \in (-\infty, 2] \cup [3, \infty) \]

2. For the product of roots:

\[ -(a^2 - 3a + 2) \geq 0 \implies a^2 - 3a + 2 \leq 0 \]

Factoring gives:

\[ (a - 1)(a - 2) \leq 0 \]

This implies:

\[ a \in [1, 2] \]

### Step 7: Find the intersection of the intervals

Now we need to find the intersection of the two conditions:

1. \( a \in (-\infty, 2] \cup [3, \infty) \)
2. \( a \in [1, 2] \)

The intersection is:

\[ a \in [1, 2] \]

### Step 8: Identify integral values

The integral values of \( a \) in the interval \( [1, 2] \) are:

\[ a = 1, 2 \]

### Conclusion

The integral values of \( a \) for which the equation has only real roots are:

\[ \boxed{1 \text{ and } 2} \]