`x^(3)-20x=x^(2)`
If a, b, and c represents the set of all values of x that satisfy the equation above, what is the value of `(a+b+c)+(abc)`?

Options

A. `-1`
B. `0`
C. `1`
D. `9`

Answer

To solve the equation \( x^3 - 20x = x^2 \), we can follow these steps:

### Step 1: Rearrange the equation
First, we rearrange the equation to bring all terms to one side:
\[ x^3 - x^2 - 20x = 0 \]

### Step 2: Identify the coefficients
Now we can identify the coefficients of the polynomial:
- The coefficient of \( x^3 \) (p) is 1.
- The coefficient of \( x^2 \) (q) is -1.
- The coefficient of \( x \) (r) is -20.
- The constant term (s) is 0.

### Step 3: Apply the relationships for roots
Using the relationships for the roots of a cubic polynomial:
1. The sum of the roots \( a + b + c = -\frac{q}{p} \)
2. The product of the roots \( abc = -\frac{s}{p} \)

### Step 4: Calculate the sum of the roots
Substituting the values of \( p \) and \( q \):
\[ a + b + c = -\frac{-1}{1} = 1 \]

### Step 5: Calculate the product of the roots
Substituting the values of \( s \) and \( p \):
\[ abc = -\frac{0}{1} = 0 \]

### Step 6: Combine the results
Now we need to find \( (a + b + c) + (abc) \):
\[ (a + b + c) + (abc) = 1 + 0 = 1 \]

### Final Answer
Thus, the value of \( (a + b + c) + (abc) \) is:
\[ \boxed{1} \]

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