If `alpha. beta` are the roots of `x^2 +bx+c=0` and `alpha + h, beta + h` are the roots of `x^2 + qx +r=0` then `2h=`

Options

A. b+q
B. b-q
C. `(b+q)/(2)`
D. `0`

Explanation

To solve the problem, we need to find the value of \(2h\) given the roots of two quadratic equations. Let's go through the solution step by step.

### Step 1: Identify the roots of the first quadratic equation
The first quadratic equation is given as:
\[ x^2 + bx + c = 0 \]
Let the roots of this equation be \(\alpha\) and \(\beta\). According to Vieta's formulas, the sum of the roots is given by:
\[ \alpha + \beta = -\frac{b}{1} = -b \]
This is our first equation.

### Step 2: Identify the roots of the second quadratic equation
The second quadratic equation is given as:
\[ x^2 + qx + r = 0 \]
The roots of this equation are \(\alpha + h\) and \(\beta + h\). Again, using Vieta's formulas, the sum of the roots for this equation is:
\[ (\alpha + h) + (\beta + h) = -\frac{q}{1} = -q \]
This can be simplified to:
\[ \alpha + \beta + 2h = -q \]
This is our second equation.

### Step 3: Substitute the value of \(\alpha + \beta\)
From our first equation, we know that \(\alpha + \beta = -b\). We can substitute this value into our second equation:
\[ -b + 2h = -q \]

### Step 4: Solve for \(2h\)
Now, we can rearrange the equation to isolate \(2h\):
\[ 2h = -q + b \]
This can also be written as:
\[ 2h = b - q \]

### Final Result
Thus, the value of \(2h\) is:
\[ \boxed{b - q} \]