If the roots of the equation `ax^2-bx-c=0` are changed by same quantity then the expression in a,b,c that does not change is

Options

A. `(b^(2)-4ac)/(a^(2))`
B. `(b - 4c)/(a)`
C. `(b^(2) + 4ac)/(a^(2))`
D. `(b^(2) - 4ac)/(a)`

Answer

To solve the problem, we need to analyze the quadratic equation \( ax^2 - bx - c = 0 \) and determine which expression involving \( a \), \( b \), and \( c \) remains unchanged when the roots are altered by the same quantity.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). According to Vieta's formulas:
\[ \alpha + \beta = \frac{b}{a} \quad \text{(Sum of roots)} \]
\[ \alpha \beta = -\frac{c}{a} \quad \text{(Product of roots)} \]

2. **Expression to Analyze**:
We want to analyze the expression \( b^2 - 4ac \). This expression is known as the discriminant of the quadratic equation.

3. **Change the Roots**:
Suppose the roots are changed by the same quantity \( n \). Thus, the new roots will be:
\[ \alpha' = \alpha - n \quad \text{and} \quad \beta' = \beta - n \]

4. **New Sum and Product of Roots**:
The new sum of the roots becomes:
\[ \alpha' + \beta' = (\alpha - n) + (\beta - n) = \alpha + \beta - 2n = \frac{b}{a} - 2n \]
The new product of the roots becomes:
\[ \alpha' \beta' = (\alpha - n)(\beta - n) = \alpha \beta - n(\alpha + \beta) + n^2 = -\frac{c}{a} - n\left(\frac{b}{a}\right) + n^2 \]

5. **Calculate the New Discriminant**:
We need to find the new discriminant based on the new roots:
\[ b'^2 - 4a c' \]
where \( b' = a(\alpha' + \beta') \) and \( c' = -a(\alpha' \beta') \).

Using the new values:
\[ b' = a\left(\frac{b}{a} - 2n\right) = b - 2an \]
and
\[ c' = -a\left(-\frac{c}{a} - n\left(\frac{b}{a}\right) + n^2\right) = c + bn - an^2 \]

6. **Substituting into the Discriminant**:
Now substituting \( b' \) and \( c' \) into the discriminant:
\[ (b - 2an)^2 - 4a(c + bn - an^2) \]

7. **Simplifying**:
Expanding this expression:
\[ (b^2 - 4ban + 4a^2n^2) - (4ac + 4abn - 4a^2n^2) \]
When we simplify, we see that the terms involving \( n \) cancel out:
\[ b^2 - 4ac \]

8. **Conclusion**:
Thus, we conclude that the expression \( b^2 - 4ac \) remains unchanged when the roots are altered by the same quantity.

### Final Answer:
The expression in \( a, b, c \) that does not change is:
\[ b^2 - 4ac \]