The square root of `14+6sqrt(5)` is

Options

A. `2+sqrt(5)`
B. `3+sqrt(5)`
C. `5+sqrt(3)`
D. `3+2sqrt(5)`

Answer

To find the square root of \( 14 + 6\sqrt{5} \), we can express it in the form of a perfect square. Let's break down the steps:

### Step 1: Assume the square root can be expressed as a binomial
We assume that:
\[ \sqrt{14 + 6\sqrt{5}} = a + b \]
where \( a \) and \( b \) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ 14 + 6\sqrt{5} = (a + b)^2 \]
Expanding the right side using the formula \( (x + y)^2 = x^2 + 2xy + y^2 \):
\[ 14 + 6\sqrt{5} = a^2 + 2ab + b^2 \]

### Step 3: Identify \( a^2 + b^2 \) and \( 2ab \)
From the equation, we can separate the rational and irrational parts:
- The rational part is \( a^2 + b^2 = 14 \)
- The irrational part is \( 2ab = 6\sqrt{5} \)

### Step 4: Solve for \( ab \)
From \( 2ab = 6\sqrt{5} \), we can simplify to find \( ab \):
\[ ab = 3\sqrt{5} \]

### Step 5: Set up a system of equations
Now we have two equations:
1. \( a^2 + b^2 = 14 \)
2. \( ab = 3\sqrt{5} \)

### Step 6: Express \( b \) in terms of \( a \)
We can express \( b \) in terms of \( a \) using the equation \( ab = 3\sqrt{5} \):
\[ b = \frac{3\sqrt{5}}{a} \]

### Step 7: Substitute \( b \) back into the first equation
Substituting \( b \) into the first equation:
\[ a^2 + \left(\frac{3\sqrt{5}}{a}\right)^2 = 14 \]
This simplifies to:
\[ a^2 + \frac{45}{a^2} = 14 \]

### Step 8: Multiply through by \( a^2 \) to eliminate the fraction
Multiplying through by \( a^2 \):
\[ a^4 - 14a^2 + 45 = 0 \]

### Step 9: Let \( x = a^2 \) and solve the quadratic
Let \( x = a^2 \), then we have:
\[ x^2 - 14x + 45 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 45}}{2 \cdot 1} \]
\[ x = \frac{14 \pm \sqrt{196 - 180}}{2} \]
\[ x = \frac{14 \pm \sqrt{16}}{2} \]
\[ x = \frac{14 \pm 4}{2} \]
Thus, we have two possible values for \( x \):
\[ x = \frac{18}{2} = 9 \quad \text{or} \quad x = \frac{10}{2} = 5 \]

### Step 10: Find \( a \) and \( b \)
1. If \( a^2 = 9 \), then \( a = 3 \).
2. If \( a^2 = 5 \), then \( a = \sqrt{5} \).

Using \( ab = 3\sqrt{5} \):
- If \( a = 3 \), then \( b = \frac{3\sqrt{5}}{3} = \sqrt{5} \).
- If \( a = \sqrt{5} \), then \( b = \frac{3\sqrt{5}}{\sqrt{5}} = 3 \).

### Conclusion
Thus, we can conclude:
\[ \sqrt{14 + 6\sqrt{5}} = 3 + \sqrt{5} \]

### Final Answer
The square root of \( 14 + 6\sqrt{5} \) is \( 3 + \sqrt{5} \).