1. Solve for the roots of the function: $$ y=x^{2}-5 x+6 $$

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To find the roots of the quadratic function:

$$
y = x^{2} - 5x + 6
$$

we need to determine the values of $$ x $$ that make $$ y = 0 $$. In other words, solve the equation:

$$
x^{2} - 5x + 6 = 0
$$

There are several methods to solve quadratic equations, such as factoring, using the quadratic formula, or completing the square. In this case, we'll use **factoring** because the quadratic can be easily factored.

### Step 1: Factor the Quadratic

We look for two numbers that multiply to the constant term $$ 6 $$ and add up to the coefficient of the middle term $$ -5 $$.

Let's list the pairs of factors of $$ 6 $$:

- $$ 1 $$ and $$ 6 $$
- $$ 2 $$ and $$ 3 $$

Now, determine which pair adds up to $$ -5 $$:

- $$ 2 + 3 = 5 $$  
  To get $$ -5 $$, both numbers should be negative: $$ -2 $$ and $$ -3 $$.

So, the quadratic can be factored as:

$$
x^{2} - 5x + 6 = (x - 2)(x - 3) = 0
$$

### Step 2: Apply the Zero Product Property

If the product of two factors is zero, at least one of the factors must be zero. Therefore:

$$
x - 2 = 0 \quad \text{or} \quad x - 3 = 0
$$

### Step 3: Solve for $$ x $$

Solving each equation:

1. $$ x - 2 = 0 $$  
   $$ x = 2 $$

2. $$ x - 3 = 0 $$  
   $$ x = 3 $$

### Conclusion

The roots of the function $$ y = x^{2} - 5x + 6 $$ are:

$$
x = 2 \quad \text{and} \quad x = 3
$$

These are the values of $$ x $$ where the quadratic function intersects the x-axis.
The roots of the equation $$ x^{2} - 5x + 6 = 0 $$ are $$ x = 2 $$ and $$ x = 3 $$.

1. Solve for the roots of the function: \[ y=x^{2}-5 x+6 \]

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