Find the distance between the zeroes of the equation: `x^(2) + 8x + 15=0`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the distance between the zeroes of the equation \(x^2 + 8x + 15 = 0\), we will follow these steps:

### Step 1: Identify the quadratic equation
The given equation is:
\[ x^2 + 8x + 15 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic equation. We are looking for two numbers that add up to 8 (the coefficient of \(x\)) and multiply to 15 (the constant term). The numbers 5 and 3 satisfy these conditions because:
\[ 5 + 3 = 8 \quad \text{and} \quad 5 \times 3 = 15 \]

So we can rewrite the equation as:
\[ x^2 + 5x + 3x + 15 = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (x^2 + 5x) + (3x + 15) = 0 \]

### Step 4: Factor by grouping
Factoring out the common terms, we get:
\[ x(x + 5) + 3(x + 5) = 0 \]
This can be factored further as:
\[ (x + 3)(x + 5) = 0 \]

### Step 5: Find the zeroes
Setting each factor to zero gives us the zeroes of the equation:
1. \(x + 3 = 0 \Rightarrow x = -3\)
2. \(x + 5 = 0 \Rightarrow x = -5\)

Thus, the zeroes of the equation are \(x = -3\) and \(x = -5\).

### Step 6: Calculate the distance between the zeroes
The distance between the zeroes can be calculated as:
\[ \text{Distance} = |(-3) - (-5)| = |-3 + 5| = |2| = 2 \]

### Final Answer
The distance between the zeroes of the equation \(x^2 + 8x + 15 = 0\) is \(2\) units.

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