If the roots of `ax^(2)+bx+c=0` are of the form `(m)/(m-1) and (m+1)/(m),` then the value of `(a+b+c)^(2)` is

Options

A. `b^(2)-2ac`
B. `2b^(2)-ac`
C. `b^(2)-4ac`
D. `2(b^(2)-2ac)`

Answer

To solve the problem, we need to find the value of \((a + b + c)^2\) given that the roots of the quadratic equation \(ax^2 + bx + c = 0\) are of the form \(\frac{m}{m-1}\) and \(\frac{m+1}{m}\).

### Step 1: Identify the roots
The roots of the equation are:
\[ r_1 = \frac{m}{m-1}, \quad r_2 = \frac{m+1}{m} \]

### Step 2: Calculate the sum of the roots
Using the formula for the sum of the roots of a quadratic equation, we have:
\[ r_1 + r_2 = -\frac{b}{a} \]
Calculating \(r_1 + r_2\):
\[ r_1 + r_2 = \frac{m}{m-1} + \frac{m+1}{m} \]
To combine these fractions, we find a common denominator:
\[ = \frac{m^2 + (m+1)(m-1)}{m(m-1)} = \frac{m^2 + (m^2 - 1)}{m(m-1)} = \frac{2m^2 - 1}{m(m-1)} \]
Thus, we have:
\[ -\frac{b}{a} = \frac{2m^2 - 1}{m(m-1)} \]

### Step 3: Calculate the product of the roots
Using the formula for the product of the roots:
\[ r_1 \cdot r_2 = \frac{c}{a} \]
Calculating \(r_1 \cdot r_2\):
\[ r_1 \cdot r_2 = \frac{m}{m-1} \cdot \frac{m+1}{m} = \frac{m+1}{m-1} \]
Thus, we have:
\[ \frac{c}{a} = \frac{m+1}{m-1} \]

### Step 4: Set up equations for \(b\) and \(c\)
From the sum of the roots:
\[ b = -a \cdot \frac{2m^2 - 1}{m(m-1)} \]
From the product of the roots:
\[ c = a \cdot \frac{m+1}{m-1} \]

### Step 5: Calculate \(a + b + c\)
Now we can express \(a + b + c\):
\[ a + b + c = a - a \cdot \frac{2m^2 - 1}{m(m-1)} + a \cdot \frac{m+1}{m-1} \]
Factoring out \(a\):
\[ = a \left(1 - \frac{2m^2 - 1}{m(m-1)} + \frac{m+1}{m-1}\right) \]

### Step 6: Simplify the expression
To simplify:
\[ = a \left(1 - \frac{2m^2 - 1}{m(m-1)} + \frac{m+1}{m-1}\right) \]
Finding a common denominator for the terms inside the parentheses:
\[ = a \left(\frac{m(m-1) - (2m^2 - 1) + (m+1)m}{m(m-1)}\right) \]
This simplifies to:
\[ = a \left(\frac{m^2 - m - 2m^2 + 1 + m^2 + m}{m(m-1)}\right) = a \left(\frac{0 + 1}{m(m-1)}\right) = \frac{a}{m(m-1)} \]

### Step 7: Calculate \((a + b + c)^2\)
Now squaring the expression:
\[ (a + b + c)^2 = \left(\frac{a}{m(m-1)}\right)^2 = \frac{a^2}{m^2(m-1)^2} \]

### Final Step: Conclusion
Since we do not have specific values for \(a\), \(b\), or \(c\), we conclude that the value of \((a + b + c)^2\) depends on the parameters of the quadratic equation and the roots given.