What is the positive square root of `[25+4sqrt(39)]` ?

Options

A. `sqrt(13)+2sqrt(3)`
B. `sqrt(13)+3sqrt(2)`
C. `sqrt(11)+2sqrt(3)`
D. `11+3sqrt(2)`

Answer

To find the positive square root of the expression \( 25 + 4\sqrt{39} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{25 + 4\sqrt{39}} \]

### Step 2: Assume a form for the square root
We assume that the square root can be expressed in the form:
\[ \sqrt{a} + \sqrt{b} \]
where \( a \) and \( b \) are numbers we need to determine.

### Step 3: Square both sides
Squaring both sides gives us:
\[ 25 + 4\sqrt{39} = (\sqrt{a} + \sqrt{b})^2 \]
Expanding the right-hand side:
\[ 25 + 4\sqrt{39} = a + b + 2\sqrt{ab} \]

### Step 4: Equate rational and irrational parts
From the equation, we can equate the rational and irrational parts:
1. \( a + b = 25 \) (rational part)
2. \( 2\sqrt{ab} = 4\sqrt{39} \) (irrational part)

### Step 5: Simplify the irrational part
From the second equation:
\[ \sqrt{ab} = 2\sqrt{39} \]
Squaring both sides gives:
\[ ab = 4 \cdot 39 = 156 \]

### Step 6: Set up a system of equations
Now we have a system of equations:
1. \( a + b = 25 \)
2. \( ab = 156 \)

### Step 7: Solve for \( a \) and \( b \)
We can use these equations to form a quadratic equation:
Let \( x \) be one of the roots, then:
\[ x^2 - (a + b)x + ab = 0 \]
Substituting the values:
\[ x^2 - 25x + 156 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{25 \pm \sqrt{25^2 - 4 \cdot 1 \cdot 156}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 25^2 - 4 \cdot 156 = 625 - 624 = 1 \]
Thus:
\[ x = \frac{25 \pm 1}{2} \]
This gives us two possible values:
\[ x = \frac{26}{2} = 13 \quad \text{and} \quad x = \frac{24}{2} = 12 \]

### Step 9: Identify \( a \) and \( b \)
Thus, we have:
\[ a = 13 \quad \text{and} \quad b = 12 \]

### Step 10: Write the final answer
Now substituting back, we find:
\[ \sqrt{25 + 4\sqrt{39}} = \sqrt{13} + \sqrt{12} \]
Since \( \sqrt{12} = 2\sqrt{3} \), we can write:
\[ \sqrt{25 + 4\sqrt{39}} = \sqrt{13} + 2\sqrt{3} \]

### Final Answer:
The positive square root of \( 25 + 4\sqrt{39} \) is:
\[ \sqrt{13} + 2\sqrt{3} \]