If the difference between the roots of `ax^(2) + bx + c = 0 ` is 1, 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)`

Explanation

To solve the problem, we need to find a relationship involving the coefficients \(a\), \(b\), and \(c\) of the quadratic equation \(ax^2 + bx + c = 0\) given that the difference between the roots is 1. Let's denote the roots of the quadratic equation as \(\alpha\) and \(\beta\).

### Step-by-Step Solution:

1. **Understanding the Roots**:
The difference between the roots is given as:
\[ \alpha - \beta = 1 \]

2. **Using Vieta's Formulas**:
According to Vieta's formulas, we know:
\[ \alpha + \beta = -\frac{b}{a} \]
\[ \alpha \beta = \frac{c}{a} \]

3. **Expressing \(\alpha\) and \(\beta\)**:
From the difference of the roots, we can express \(\alpha\) in terms of \(\beta\):
\[ \alpha = \beta + 1 \]

4. **Substituting into Vieta's Formula**:
Substitute \(\alpha\) into the sum of the roots:
\[ (\beta + 1) + \beta = -\frac{b}{a} \]
Simplifying this gives:
\[ 2\beta + 1 = -\frac{b}{a} \]
Rearranging, we find:
\[ 2\beta = -\frac{b}{a} - 1 \quad \Rightarrow \quad \beta = -\frac{b}{2a} - \frac{1}{2} \]

5. **Finding \(\alpha\)**:
Now, substituting \(\beta\) back to find \(\alpha\):
\[ \alpha = \beta + 1 = \left(-\frac{b}{2a} - \frac{1}{2}\right) + 1 = -\frac{b}{2a} + \frac{1}{2} \]

6. **Calculating \(\alpha \beta\)**:
Now we can calculate \(\alpha \beta\):
\[ \alpha \beta = \left(-\frac{b}{2a} + \frac{1}{2}\right)\left(-\frac{b}{2a} - \frac{1}{2}\right) \]
This is a difference of squares:
\[ \alpha \beta = \left(-\frac{b}{2a}\right)^2 - \left(\frac{1}{2}\right)^2 = \frac{b^2}{4a^2} - \frac{1}{4} \]

7. **Using Vieta's Formula for Product of Roots**:
According to Vieta's, we also know:
\[ \alpha \beta = \frac{c}{a} \]
Setting the two expressions for \(\alpha \beta\) equal gives:
\[ \frac{b^2}{4a^2} - \frac{1}{4} = \frac{c}{a} \]

8. **Clearing the Denominator**:
Multiply through by \(4a^2\):
\[ b^2 - a^2 = 4ac \]

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

### Conclusion:
Thus, the correct relationship that holds when the difference between the roots of the quadratic equation \(ax^2 + bx + c = 0\) is 1 is:
\[ b^2 = a(a + 4c) \]