Given that `d/4 + 8/d + 3 = 0`, what is d?

Answer

To solve the equation \( \frac{d}{4} + \frac{8}{d} + 3 = 0 \), we will follow these steps:

### Step 1: Clear the fractions
To eliminate the fractions, we can multiply the entire equation by \( 4d \) (the least common multiple of the denominators):

\[ 4d \left( \frac{d}{4} \right) + 4d \left( \frac{8}{d} \right) + 4d(3) = 0 \]

This simplifies to:

\[ d^2 + 32 + 12d = 0 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to standard quadratic form:

\[ d^2 + 12d + 32 = 0 \]

### Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation. We need two numbers that multiply to \( 32 \) and add up to \( 12 \). The numbers are \( 4 \) and \( 8 \):

\[ (d + 4)(d + 8) = 0 \]

### Step 4: Solve for \( d \)
Now, we can set each factor equal to zero:

1. \( d + 4 = 0 \) → \( d = -4 \)
2. \( d + 8 = 0 \) → \( d = -8 \)

Thus, the solutions for \( d \) are:

\[ d = -4 \quad \text{or} \quad d = -8 \]

### Final Answer
The values of \( d \) are \( -4 \) and \( -8 \).

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