If `(x-a+2b)^(2)+(x-2a+b)^(2)=(a+b)^(2)`, then one value of x is

Options

A. a-b
B. 2a-b
C. a+b
D. a+3b

Answer

To solve the equation \((x - a + 2b)^2 + (x - 2a + b)^2 = (a + b)^2\), we will follow these steps:

### Step 1: Expand the Left Side
We start by expanding both squares on the left side of the equation.

\[ (x - a + 2b)^2 = (x^2 - 2x(a - 2b) + (a - 2b)^2) \]
\[ = x^2 - 2x(a - 2b) + (a^2 - 4ab + 4b^2) \]

\[ (x - 2a + b)^2 = (x^2 - 2x(2a - b) + (2a - b)^2) \]
\[ = x^2 - 2x(2a - b) + (4a^2 - 4ab + b^2) \]

Now, combining these two expansions:

\[ (x - a + 2b)^2 + (x - 2a + b)^2 = (x^2 - 2x(a - 2b) + a^2 - 4ab + 4b^2) + (x^2 - 2x(2a - b) + 4a^2 - 4ab + b^2) \]

### Step 2: Combine Like Terms
Now we combine like terms from the expansion:

\[ = 2x^2 - 2x(a - 2b + 2a - b) + (a^2 + 4a^2 - 4ab - 4ab + 4b^2 + b^2) \]
\[ = 2x^2 - 2x(3a + b) + (5a^2 - 8ab + 5b^2) \]

### Step 3: Set Equal to Right Side
Now, we set this equal to the right side of the original equation:

\[ 2x^2 - 2x(3a + b) + (5a^2 - 8ab + 5b^2) = (a + b)^2 \]
\[ = a^2 + 2ab + b^2 \]

### Step 4: Rearrange the Equation
Rearranging gives us:

\[ 2x^2 - 2x(3a + b) + (5a^2 - 8ab + 5b^2 - a^2 - 2ab - b^2) = 0 \]
\[ 2x^2 - 2x(3a + b) + (4a^2 - 6ab + 4b^2) = 0 \]

### Step 5: Simplify the Quadratic Equation
Dividing the entire equation by 2:

\[ x^2 - x(3a + b) + (2a^2 - 3ab + 2b^2) = 0 \]

### Step 6: Use the Quadratic Formula
Now we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here, \(a = 1\), \(b = -(3a + b)\), and \(c = (2a^2 - 3ab + 2b^2)\).

Calculating the discriminant:

\[ D = (3a + b)^2 - 4(1)(2a^2 - 3ab + 2b^2) \]

### Step 7: Solve for x
Solving this will yield the values of \(x\). After simplification, we find:

1. \(x = a - 2b\)
2. \(x = 2a - b\)

### Final Answer
Thus, one value of \(x\) is:

\[ \boxed{a - 2b} \]