solve `x^(2)+8x+15=0`

Answer

To solve the quadratic equation \(x^2 + 8x + 15 = 0\), we can use the method of middle term factorization. Here’s a step-by-step solution:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Here, we have:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 8\) (coefficient of \(x\))
- \(c = 15\) (constant term)

### Step 2: Find two numbers that add up to \(b\) and multiply to \(c\)
We need to find two numbers that:
- Add up to \(8\) (the coefficient \(b\))
- Multiply to \(15\) (the constant \(c\))

The numbers that satisfy these conditions are \(5\) and \(3\):
- \(5 + 3 = 8\)
- \(5 \times 3 = 15\)

### Step 3: Rewrite the middle term
We can rewrite the equation by splitting the middle term \(8x\) into \(5x + 3x\):
\[ x^2 + 5x + 3x + 15 = 0 \]

### Step 4: Factor the equation
Now, we can group the terms:
\[ (x^2 + 5x) + (3x + 15) = 0 \]
Factoring out the common terms:
\[ x(x + 5) + 3(x + 5) = 0 \]
Now, factor out the common factor \((x + 5)\):
\[ (x + 5)(x + 3) = 0 \]

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

### Step 6: Solve for \(x\)
Solving these equations gives:
1. \(x = -5\)
2. \(x = -3\)

### Final Answer
The solutions to the equation \(x^2 + 8x + 15 = 0\) are:
\[ x = -5 \quad \text{and} \quad x = -3 \]

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