If the sum of the roots of the equation `ax^(2) + bx + c =0` is equal to the sum of their squares. Then:

Options

A. A) `a^(2) + b^(2) = c^(2)`
B. B) `a^(2) + b^(2) = a + b`
C. C) `ab + b^(2) = 2ac`
D. D) `ab - b^(2) = 2ac`

Answer

To solve the problem, we need to establish the relationship between the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \) given that the sum of the roots is equal to the sum of their squares.

### Step-by-step Solution:

1. **Identify the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \).

2. **Sum of the Roots**:
The sum of the roots can be expressed using Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} \]

3. **Sum of the Squares of the Roots**:
The sum of the squares of the roots can be expressed as:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Using Vieta's formulas again, we know that:
\[ \alpha\beta = \frac{c}{a} \]
Therefore, we can rewrite the sum of the squares as:
\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) \]
Simplifying this gives:
\[ \alpha^2 + \beta^2 = \frac{b^2}{a^2} - \frac{2c}{a} \]

4. **Set the Two Expressions Equal**:
According to the problem, the sum of the roots is equal to the sum of their squares:
\[ -\frac{b}{a} = \frac{b^2}{a^2} - \frac{2c}{a} \]

5. **Clear the Denominators**:
Multiply through by \( a^2 \) to eliminate the denominators:
\[ -ba = b^2 - 2ac \]

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

7. **Final Expression**:
Thus, we have derived the relationship:
\[ b^2 + ab - 2ac = 0 \]

### Conclusion:
The relationship that holds true when the sum of the roots is equal to the sum of their squares is:
\[ b^2 + ab = 2ac \]