Solve each of the following quadratic equations:
`x^(2)-(2b-1)x+(b^(2)-b-20)=0`

Explanation

To solve the quadratic equation \( x^2 - (2b - 1)x + (b^2 - b - 20) = 0 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = -(2b - 1) = -2b + 1 \)
- \( c = b^2 - b - 20 \)

### Step 2: Use the quadratic formula
The quadratic formula to find the roots of the equation \( ax^2 + bx + c = 0 \) is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-2b + 1) \pm \sqrt{(-2b + 1)^2 - 4 \cdot 1 \cdot (b^2 - b - 20)}}{2 \cdot 1} \]

### Step 3: Simplify the expression
First, simplify \( -(-2b + 1) \):
\[ x = \frac{2b - 1 \pm \sqrt{(-2b + 1)^2 - 4(b^2 - b - 20)}}{2} \]
Now, calculate \( (-2b + 1)^2 \):
\[ (-2b + 1)^2 = 4b^2 - 4b + 1 \]
Next, calculate \( 4(b^2 - b - 20) \):
\[ 4(b^2 - b - 20) = 4b^2 - 4b - 80 \]
Now, substitute these into the equation:
\[ x = \frac{2b - 1 \pm \sqrt{(4b^2 - 4b + 1) - (4b^2 - 4b - 80)}}{2} \]
This simplifies to:
\[ x = \frac{2b - 1 \pm \sqrt{1 + 80}}{2} \]
\[ x = \frac{2b - 1 \pm \sqrt{81}}{2} \]
\[ x = \frac{2b - 1 \pm 9}{2} \]

### Step 4: Find the roots
Now, we can find the two possible values for \( x \):
1. \( x = \frac{2b - 1 + 9}{2} = \frac{2b + 8}{2} = b + 4 \)
2. \( x = \frac{2b - 1 - 9}{2} = \frac{2b - 10}{2} = b - 5 \)

### Final Solution
Thus, the solutions to the quadratic equation \( x^2 - (2b - 1)x + (b^2 - b - 20) = 0 \) are:
\[ x = b + 4 \quad \text{and} \quad x = b - 5 \]

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