If one of the roots of the equation `x^(2) -bx + c = 0` is the square of the other, then which of the following option is correct ?

Options

A. `b^(3) =3bc + c^(2) + c`
B. `c^(3) = 3bc + b^(2) +b`
C. `3bc = c^(3) + b^(2) +b`
D. `3 "fac" =c^(3) + b^(3) + b^(2)`

Answer

To solve the problem step by step, we need to determine the relationship between the roots of the quadratic equation \(x^2 - bx + c = 0\) given that one root is the square of the other. Let's denote the roots as \(\alpha\) and \(\beta\), where \(\beta = \alpha^2\).

### Step 1: Write down the relationship between the roots
According to Vieta's formulas, for the quadratic equation \(x^2 - bx + c = 0\):
- The sum of the roots \(\alpha + \beta = b\)
- The product of the roots \(\alpha \cdot \beta = c\)

### Step 2: Substitute \(\beta\) with \(\alpha^2\)
Since we know that \(\beta = \alpha^2\), we can substitute this into the equations:
- The sum of the roots becomes:
\[ \alpha + \alpha^2 = b \]
- The product of the roots becomes:
\[ \alpha \cdot \alpha^2 = \alpha^3 = c \]

### Step 3: Rearranging the equations
From the sum equation:
\[ \alpha^2 + \alpha - b = 0 \]
From the product equation:
\[ \alpha^3 - c = 0 \]

### Step 4: Express \(b\) and \(c\) in terms of \(\alpha\)
From the first equation, we can express \(b\):
\[ b = \alpha + \alpha^2 \]
From the second equation, we can express \(c\):
\[ c = \alpha^3 \]

### Step 5: Substitute \(b\) and \(c\) into the derived equation
Now, we can analyze the relationship between \(b\) and \(c\):
1. Substitute \(b\) and \(c\) into the equation \(b^3 = 3bc + c^2\):
\[ (\alpha + \alpha^2)^3 = 3(\alpha + \alpha^2)(\alpha^3) + (\alpha^3)^2 \]

### Step 6: Simplifying the equation
Now we expand and simplify both sides:
- Left-hand side:
\[ (\alpha + \alpha^2)^3 = \alpha^3 + 3\alpha^2(\alpha^2) + 3\alpha(\alpha^2)^2 + \alpha^6 = \alpha^3 + 3\alpha^4 + 3\alpha^5 + \alpha^6 \]
- Right-hand side:
\[ 3(\alpha + \alpha^2)(\alpha^3) + (\alpha^3)^2 = 3(\alpha^4 + \alpha^5) + \alpha^6 = 3\alpha^4 + 3\alpha^5 + \alpha^6 \]

### Step 7: Equating both sides
Now we equate the left-hand side and right-hand side:
\[ \alpha^3 + 3\alpha^4 + 3\alpha^5 + \alpha^6 = 3\alpha^4 + 3\alpha^5 + \alpha^6 \]
This simplifies to:
\[ \alpha^3 = 0 \]
Thus, \(\alpha = 0\) or \(\alpha\) can take other values.

### Conclusion
We have derived the conditions for \(b\) and \(c\) based on the roots of the quadratic equation. The correct option can be determined by substituting specific values of \(b\) and \(c\) into the original equation.