Find the roots of the equation `x^2+7x+10=0`.

Options

A. `-2,-5`
B. `1,-5`
C. `-2,3`
D. `1,3`

Answer

To find the roots of the quadratic equation \(x^2 + 7x + 10 = 0\), we will use the factorization method. Here are the steps:

### Step 1: Write the equation
We start with the given equation:
\[ x^2 + 7x + 10 = 0 \]

### Step 2: Factor the quadratic expression
We need to factor the quadratic expression \(x^2 + 7x + 10\). We look for two numbers that multiply to \(10\) (the constant term) and add up to \(7\) (the coefficient of \(x\)). The numbers \(2\) and \(5\) satisfy these conditions because:
\[ 2 \times 5 = 10 \quad \text{and} \quad 2 + 5 = 7 \]
Thus, we can rewrite the equation as:
\[ x^2 + 2x + 5x + 10 = 0 \]

### Step 3: Group the terms
Next, we group the terms:
\[ (x^2 + 2x) + (5x + 10) = 0 \]

### Step 4: Factor by grouping
Now we factor out the common factors in each group:
\[ x(x + 2) + 5(x + 2) = 0 \]
We can factor out \((x + 2)\):
\[ (x + 2)(x + 5) = 0 \]

### Step 5: Solve for the roots
Now, we set each factor equal to zero:
1. \(x + 2 = 0\) leads to \(x = -2\)
2. \(x + 5 = 0\) leads to \(x = -5\)

### Conclusion
The roots of the equation \(x^2 + 7x + 10 = 0\) are:
\[ x = -2 \quad \text{and} \quad x = -5 \]