Given that the equation `3x^2+2x-8=0` has two distinct solutions, what is the value of the smaller solution subtrated from the larger solution?

Answer

To solve the quadratic equation \(3x^2 + 2x - 8 = 0\) and find the difference between the larger and smaller solutions, we can follow these steps:

### Step 1: Identify the quadratic equation
The given equation is:
\[ 3x^2 + 2x - 8 = 0 \]

### Step 2: Use the quadratic formula to find the solutions
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 3\), \(b = 2\), and \(c = -8\).

### Step 3: Calculate the discriminant
First, we calculate the discriminant \(D = b^2 - 4ac\):
\[ D = 2^2 - 4 \cdot 3 \cdot (-8) = 4 + 96 = 100 \]

### Step 4: Calculate the solutions using the quadratic formula
Now substituting the values into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{100}}{2 \cdot 3} = \frac{-2 \pm 10}{6} \]

This gives us two solutions:
1. For the positive root:
\[ x_1 = \frac{-2 + 10}{6} = \frac{8}{6} = \frac{4}{3} \]
2. For the negative root:
\[ x_2 = \frac{-2 - 10}{6} = \frac{-12}{6} = -2 \]

### Step 5: Identify the larger and smaller solutions
The two solutions are:
- \(x_1 = \frac{4}{3}\) (larger solution)
- \(x_2 = -2\) (smaller solution)

### Step 6: Find the difference between the larger and smaller solutions
We need to find the value of the larger solution subtracted by the smaller solution:
\[ \text{Difference} = x_1 - x_2 = \frac{4}{3} - (-2) = \frac{4}{3} + 2 \]

### Step 7: Simplify the expression
Convert 2 into a fraction with a common denominator:
\[ 2 = \frac{6}{3} \]
So,
\[ \text{Difference} = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \]

### Conclusion
The value of the smaller solution subtracted from the larger solution is:
\[ \frac{10}{3} \]