`(3k+14)k=5`
If r and s represents the solutions of the equation above and `rgts`, what is the value of the difference r-s?

Answer

To solve the equation \((3k + 14)k = 5\) and find the difference \(r - s\) where \(r\) and \(s\) are the solutions, we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ (3k + 14)k = 5 \]
Expanding the left side gives:
\[ 3k^2 + 14k = 5 \]
Now, we rearrange it to set it to zero:
\[ 3k^2 + 14k - 5 = 0 \]

### Step 2: Identify coefficients
In the standard quadratic form \(ax^2 + bx + c = 0\), we identify:
- \(a = 3\)
- \(b = 14\)
- \(c = -5\)

### Step 3: Use the quadratic formula
The quadratic formula to find the roots \(k\) is given by:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ k = \frac{-14 \pm \sqrt{14^2 - 4 \cdot 3 \cdot (-5)}}{2 \cdot 3} \]

### Step 4: Calculate the discriminant
Calculating the discriminant:
\[ b^2 - 4ac = 196 + 60 = 256 \]
So we have:
\[ k = \frac{-14 \pm \sqrt{256}}{6} \]

### Step 5: Simplify the square root
Calculating the square root:
\[ \sqrt{256} = 16 \]
Thus, the equation becomes:
\[ k = \frac{-14 \pm 16}{6} \]

### Step 6: Calculate the two solutions
Now we calculate the two possible values for \(k\):
1. \(k_1 = \frac{-14 + 16}{6} = \frac{2}{6} = \frac{1}{3}\)
2. \(k_2 = \frac{-14 - 16}{6} = \frac{-30}{6} = -5\)

### Step 7: Identify \(r\) and \(s\)
Let \(r = \frac{1}{3}\) and \(s = -5\) (since \(r > s\)).

### Step 8: Calculate the difference \(r - s\)
Now we find the difference:
\[ r - s = \frac{1}{3} - (-5) = \frac{1}{3} + 5 = \frac{1}{3} + \frac{15}{3} = \frac{16}{3} \]

### Final Answer
Thus, the value of \(r - s\) is:
\[ \frac{16}{3} \]

---