If the solution to `2x^(2)-8x-5=0` are p and q with `pgtq`, what is the value of p-q?

Options

A. `sqrt(26)`
B. `(7)/(2)`
C. `2sqrt(13)`
D. `(11)/(2)`

Answer

To solve the quadratic equation \(2x^2 - 8x - 5 = 0\) and find the value of \(p - q\) where \(p\) and \(q\) are the roots of the equation with \(p > q\), we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \(ax^2 + bx + c = 0\). Here, we have:
- \(a = 2\)
- \(b = -8\)
- \(c = -5\)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas, we know:
- The sum of the roots \(p + q = -\frac{b}{a} = -\frac{-8}{2} = \frac{8}{2} = 4\)
- The product of the roots \(pq = \frac{c}{a} = \frac{-5}{2}\)

### Step 3: Use the identity to find \(p - q\)
We can use the identity:
\[ p - q = \sqrt{(p + q)^2 - 4pq} \]
Substituting the values we found:
\[ p - q = \sqrt{(4)^2 - 4\left(-\frac{5}{2}\right)} \]

### Step 4: Simplify the expression
Calculating inside the square root:
\[ p - q = \sqrt{16 + 10} = \sqrt{26} \]

### Step 5: Determine the final value
Since \(p > q\), we take the positive root:
\[ p - q = \sqrt{26} \]

Thus, the value of \(p - q\) is \(\sqrt{26}\).

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