What is the value of positive square root of 69 + `28sqrt()5`?

Options

A. `7+ 2sqrt()5`
B. `7- 2sqrt()5`
C. `2+ 7sqrt()5`
D. `2- 7sqrt()5`

Answer

To find the positive square root of \( 69 + 28\sqrt{5} \), we can follow these steps:

### Step 1: Set up the expression
We need to find the square root of the expression:
\[ \sqrt{69 + 28\sqrt{5}} \]

### Step 2: Assume the square root can be expressed in a specific form
We assume that the expression can be written in the form:
\[ \sqrt{a} + \sqrt{b} \]
where \( a \) and \( b \) are to be determined.

### Step 3: Square both sides
Squaring both sides gives us:
\[ 69 + 28\sqrt{5} = (\sqrt{a} + \sqrt{b})^2 \]
Expanding the right side using the formula \( (x + y)^2 = x^2 + y^2 + 2xy \):
\[ 69 + 28\sqrt{5} = a + b + 2\sqrt{ab} \]

### Step 4: Equate the rational and irrational parts
From this equation, we can equate the rational parts and the coefficients of the square root:
1. \( a + b = 69 \)
2. \( 2\sqrt{ab} = 28\sqrt{5} \)

### Step 5: Solve for \( ab \)
From the second equation, we can isolate \( \sqrt{ab} \):
\[ \sqrt{ab} = 14\sqrt{5} \]
Squaring both sides gives:
\[ ab = 196 \cdot 5 = 980 \]

### Step 6: Set up a system of equations
Now we have a system of two equations:
1. \( a + b = 69 \)
2. \( ab = 980 \)

### Step 7: Solve the quadratic equation
We can express \( a \) and \( b \) as the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \]
Substituting the known values:
\[ x^2 - 69x + 980 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{69 \pm \sqrt{69^2 - 4 \cdot 1 \cdot 980}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 69^2 = 4761 \]
\[ 4 \cdot 980 = 3920 \]
\[ 69^2 - 4 \cdot 980 = 4761 - 3920 = 841 \]
Now substituting back:
\[ x = \frac{69 \pm \sqrt{841}}{2} \]
Since \( \sqrt{841} = 29 \):
\[ x = \frac{69 \pm 29}{2} \]
Calculating the two possible values:
1. \( x = \frac{98}{2} = 49 \)
2. \( x = \frac{40}{2} = 20 \)

### Step 9: Identify \( a \) and \( b \)
Thus, we have \( a = 49 \) and \( b = 20 \).

### Step 10: Write the final expression
Now we can express the square root:
\[ \sqrt{69 + 28\sqrt{5}} = \sqrt{49} + \sqrt{20} = 7 + 2\sqrt{5} \]

### Final Answer
The positive square root of \( 69 + 28\sqrt{5} \) is:
\[ \boxed{7 + 2\sqrt{5}} \]