If the difference between the roots of `ax^(2) + bx +c = 0` is I. then which one of the following is correct?

Options

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

Answer

To solve the problem, we need to analyze the quadratic equation given as \( ax^2 + bx + c = 0 \) and find the relationship between the coefficients and the roots based on the information that the difference between the roots is 1.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the quadratic equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \beta \). We are given that the difference between the roots is \( \alpha - \beta = 1 \).

2. **Square the Difference**:
To eliminate the square root, we square both sides of the equation:
\[ (\alpha - \beta)^2 = 1^2 \]
This gives us:
\[ \alpha^2 - 2\alpha\beta + \beta^2 = 1 \]

3. **Use the Sum and Product of Roots**:
We know from Vieta's formulas that:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

We can express \( \alpha^2 + \beta^2 \) in terms of the sum and product:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values:
\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) \]

4. **Substitute into the Equation**:
Now we substitute this back into our squared difference equation:
\[ \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) - 2\left(\frac{c}{a}\right) = 1 \]
This simplifies to:
\[ \frac{b^2}{a^2} - \frac{4c}{a} = 1 \]

5. **Clear the Denominator**:
Multiply through by \( a^2 \) to eliminate the fractions:
\[ b^2 - 4ac = a^2 \]

6. **Rearrange the Equation**:
Rearranging gives us:
\[ b^2 = a^2 + 4ac \]

7. **Factor the Right Side**:
We can factor the right side:
\[ b^2 = a(a + 4c) \]

### Conclusion:
From the above steps, we find that the correct relationship is:
\[ b^2 = a(a + 4c) \]
Thus, the correct option is **Option A**: \( b^2 = a(a + 4c) \).