Find the roots of the following equations.
`x^(2)-7x+12=0`

Answer

To find the roots of the quadratic equation \( x^2 - 7x + 12 = 0 \), we can use the middle term factorization method. Here are the steps to solve the equation:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -7 \)
- \( c = 12 \)

### Step 2: Factor the quadratic expression
We need to factor the quadratic expression \( x^2 - 7x + 12 \). We look for two numbers that multiply to \( c \) (which is 12) and add up to \( b \) (which is -7).

The two numbers that satisfy these conditions are -3 and -4, since:
- \( -3 \times -4 = 12 \)
- \( -3 + -4 = -7 \)

Thus, we can rewrite the equation as:
\[ x^2 - 3x - 4x + 12 = 0 \]

### Step 3: Group the terms
Now, we group the terms:
\[ (x^2 - 3x) + (-4x + 12) = 0 \]

### Step 4: Factor by grouping
We can factor out the common factors in each group:
\[ x(x - 3) - 4(x - 3) = 0 \]

Now, we can factor out \( (x - 3) \):
\[ (x - 3)(x - 4) = 0 \]

### Step 5: Set each factor to zero
Now we set each factor equal to zero:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 4 = 0 \) → \( x = 4 \)

### Conclusion
The roots of the equation \( x^2 - 7x + 12 = 0 \) are:
\[ x = 3 \quad \text{and} \quad x = 4 \]