If `4x^(2) + y^(2)=a and xy= b`, find the value of `2x+y`

Answer

To solve the problem, we need to find the value of \(2x + y\) given the equations \(4x^2 + y^2 = a\) and \(xy = b\).

### Step-by-Step Solution:

1. **Start with the expression**: We want to find \(2x + y\). To do this, we will square the expression:
\[ (2x + y)^2 \]

2. **Apply the square identity**: Using the identity \((a + b)^2 = a^2 + b^2 + 2ab\), we can expand \((2x + y)^2\):
\[ (2x + y)^2 = (2x)^2 + y^2 + 2(2x)(y) \]
This simplifies to:
\[ = 4x^2 + y^2 + 4xy \]

3. **Substitute the known values**: We know from the problem statement that \(4x^2 + y^2 = a\) and \(xy = b\). Therefore, we can substitute these values into our expression:
\[ (2x + y)^2 = 4x^2 + y^2 + 4xy = a + 4b \]

4. **Take the square root**: To find \(2x + y\), we take the square root of both sides:
\[ 2x + y = \sqrt{a + 4b} \]

### Final Answer:
Thus, the value of \(2x + y\) is:
\[ \boxed{\sqrt{a + 4b}} \]