If ` ax^(2) + bx + c = a (x - p)^(2) ` , then the relation among a, b, c would be

Options

A. abc = 1
B. ` b^(2) = ac`
C. `b^(2) =4ac`
D. `2 b = a + c`

Answer

To solve the equation \( ax^2 + bx + c = a(x - p)^2 \) and find the relation among \( a, b, \) and \( c \), we can follow these steps:

### Step 1: Expand the right-hand side
We start with the equation:
\[ ax^2 + bx + c = a(x - p)^2 \]
Expanding the right-hand side:
\[ a(x - p)^2 = a(x^2 - 2px + p^2) = ax^2 - 2apx + ap^2 \]
So, we can rewrite the equation as:
\[ ax^2 + bx + c = ax^2 - 2apx + ap^2 \]

### Step 2: Compare coefficients
Since the left-hand side and right-hand side must be equal for all \( x \), we can compare the coefficients of \( x^2 \), \( x \), and the constant term.

1. Coefficient of \( x^2 \):
\[ a = a \quad \text{(This is always true)} \]

2. Coefficient of \( x \):
\[ b = -2ap \]

3. Constant term:
\[ c = ap^2 \]

### Step 3: Express \( p^2 \) in terms of \( c \) and \( a \)
From the equation \( c = ap^2 \), we can express \( p^2 \) as:
\[ p^2 = \frac{c}{a} \]

### Step 4: Substitute \( p^2 \) into the equation for \( b \)
Now, we substitute \( p^2 \) into the equation \( b = -2ap \). First, we need to express \( p \) in terms of \( c \) and \( a \):
\[ p = \sqrt{\frac{c}{a}} \]
Substituting this into the equation for \( b \):
\[ b = -2a\left(\sqrt{\frac{c}{a}}\right) \]
Simplifying this gives:
\[ b = -2\sqrt{ac} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides:
\[ b^2 = 4ac \]

### Conclusion
The relation among \( a, b, \) and \( c \) is:
\[ b^2 = 4ac \]