The roots of the quadratic polynomial `x^2-5x+6=0` is:

Options

A. `-2, -3`
B. `-2, 3`
C. `3, -2`
D. 3, 2

Answer

To find the roots of the quadratic polynomial \(x^2 - 5x + 6 = 0\), we can follow these steps:

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ x^2 - 5x + 6 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic equation. We are looking for two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)). The numbers that satisfy this condition are \(-2\) and \(-3\).

So we can rewrite the equation as:
\[ x^2 - 2x - 3x + 6 = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (x^2 - 2x) + (-3x + 6) = 0 \]

### Step 4: Factor by grouping
Now, we factor out the common terms from each group:
\[ x(x - 2) - 3(x - 2) = 0 \]

### Step 5: Factor out the common binomial
Now we can factor out the common binomial \((x - 2)\):
\[ (x - 2)(x - 3) = 0 \]

### Step 6: Set each factor to zero
To find the roots, we set each factor equal to zero:
1. \(x - 2 = 0 \implies x = 2\)
2. \(x - 3 = 0 \implies x = 3\)

### Conclusion
The roots of the quadratic polynomial \(x^2 - 5x + 6 = 0\) are:
\[ x = 2 \quad \text{and} \quad x = 3 \]

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