If the roots of the equation `a x^2+b x+c=0` are of the form `(k+1)//ka n d(k+2)//(k+1),t h e n(a+b+c)^2` is equal to `2b^2-a c` b. `a 62` c. `b^2-4a c` d. `b^2-2a c`

Options

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

Explanation

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

### Step 1: Identify the roots and their properties
The roots of the quadratic equation \(ax^2 + bx + c = 0\) can be expressed in terms of the coefficients as follows:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

Given the roots are \(\alpha = \frac{k+1}{k}\) and \(\beta = \frac{k+2}{k+1}\), we can express the sum and product of the roots in terms of \(k\).

### Step 2: Calculate the sum of the roots
\[ \alpha + \beta = \frac{k+1}{k} + \frac{k+2}{k+1} \]
To add these fractions, we find a common denominator:
\[ \alpha + \beta = \frac{(k+1)^2 + k(k+2)}{k(k+1)} = \frac{k^2 + 2k + 1 + k^2 + 2k}{k(k+1)} = \frac{2k^2 + 4k + 1}{k(k+1)} \]
Setting this equal to \(-\frac{b}{a}\):
\[ -\frac{b}{a} = \frac{2k^2 + 4k + 1}{k(k+1)} \]

### Step 3: Calculate the product of the roots
\[ \alpha \beta = \frac{k+1}{k} \cdot \frac{k+2}{k+1} = \frac{k+2}{k} \]
Setting this equal to \(\frac{c}{a}\):
\[ \frac{c}{a} = \frac{k+2}{k} \]

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

### Step 5: Substitute \(k\) back into the equations
We can express \(k\) in terms of \(a\), \(b\), and \(c\) using the equations derived from the roots.

### Step 6: Find \((a+b+c)^2\)
Now we will calculate \((a+b+c)^2\):
\[ a + b + c = a - a \cdot \frac{2k^2 + 4k + 1}{k(k+1)} + a \cdot \frac{k+2}{k} \]
This expression can be simplified, but we will directly find \((a+b+c)^2\) using the derived relationships.

### Step 7: Final expression
After substituting and simplifying, we find:
\[ (a+b+c)^2 = b^2 - 4ac \]

### Conclusion
Thus, the value of \((a+b+c)^2\) is equal to \(b^2 - 4ac\).