What is the positive difference between the x-intercepts of the parabola given by the equatin `g (x) =-2.5 x ^(2) +10x -7.5` ?

Explanation

To find the positive difference between the x-intercepts of the parabola given by the equation \( g(x) = -2.5x^2 + 10x - 7.5 \), we will follow these steps:

### Step 1: Set the equation equal to zero
To find the x-intercepts, we need to set \( g(x) = 0 \):
\[ -2.5x^2 + 10x - 7.5 = 0 \]

### Step 2: Simplify the equation
We can simplify the equation by dividing all terms by -2.5:
\[ x^2 - 4x + 3 = 0 \]

### Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation \( x^2 - 4x + 3 \):
\[ (x - 1)(x - 3) = 0 \]

### Step 4: Solve for x
Setting each factor equal to zero gives us the x-intercepts:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Step 5: Find the positive difference between the x-intercepts
Now, we find the positive difference between the x-intercepts \( x = 1 \) and \( x = 3 \):
\[ \text{Positive difference} = 3 - 1 = 2 \]

### Final Answer
The positive difference between the x-intercepts is \( 2 \).

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