Solve each of the following equatins :
`x^(2)+12x+35=0`

Answer

To solve the quadratic equation \( x^2 + 12x + 35 = 0 \) using the method of factorization, we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 12 \)
- \( c = 35 \)

### Step 2: Calculate \( a \times c \)
We need to find the product \( a \times c \):
\[ a \times c = 1 \times 35 = 35 \]

### Step 3: Find two numbers that add up to \( b \) and multiply to \( a \times c \)
We need to find two numbers that add up to \( b = 12 \) and multiply to \( a \times c = 35 \). The numbers that satisfy these conditions are \( 7 \) and \( 5 \) because:
\[ 7 + 5 = 12 \quad \text{and} \quad 7 \times 5 = 35 \]

### Step 4: Rewrite the equation
We can rewrite the middle term \( 12x \) using the two numbers we found:
\[ x^2 + 7x + 5x + 35 = 0 \]

### Step 5: Factor by grouping
Now, we can group the terms:
\[ (x^2 + 7x) + (5x + 35) = 0 \]
Factoring out the common terms in each group gives us:
\[ x(x + 7) + 5(x + 7) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \( (x + 7) \):
\[ (x + 7)(x + 5) = 0 \]

### Step 7: Set each factor to zero
Now, we set each factor equal to zero to find the solutions:
1. \( x + 7 = 0 \) → \( x = -7 \)
2. \( x + 5 = 0 \) → \( x = -5 \)

### Conclusion
The solutions to the equation \( x^2 + 12x + 35 = 0 \) are:
\[ x = -7 \quad \text{and} \quad x = -5 \]

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