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 the expression \(69 + 28\sqrt{5}\), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \sqrt{69 + 28\sqrt{5}} \]

### Step 2: Assume the Form of the Square Root
We assume that the square root can be expressed in the form:
\[ \sqrt{a} + \sqrt{b} \]
where \(a\) and \(b\) are positive numbers.

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

### Step 4: Equate the Rational and Irrational Parts
From the equation \(69 + 28\sqrt{5} = a + b + 2\sqrt{ab}\), we can equate the rational and irrational parts:
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 equations:
1. \(a + b = 69\)
2. \(ab = 980\)

### Step 7: Solve the Quadratic Equation
We can express \(b\) in terms of \(a\):
\[ b = 69 - a \]
Substituting into the second equation:
\[ a(69 - a) = 980 \]
This simplifies to:
\[ 69a - a^2 = 980 \]
Rearranging gives us:
\[ a^2 - 69a + 980 = 0 \]

### Step 8: Use the Quadratic Formula
We can solve for \(a\) using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(b = -69\), \(a = 1\), and \(c = 980\):
\[ a = \frac{69 \pm \sqrt{69^2 - 4 \cdot 1 \cdot 980}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 69^2 = 4761 \quad \text{and} \quad 4 \cdot 980 = 3920 \]
So:
\[ 69^2 - 4 \cdot 980 = 4761 - 3920 = 841 \]
Thus:
\[ a = \frac{69 \pm \sqrt{841}}{2} = \frac{69 \pm 29}{2} \]

### Step 9: Calculate Values of \(a\)
Calculating the two possible values:
1. \(a = \frac{98}{2} = 49\)
2. \(a = \frac{40}{2} = 20\)

### Step 10: Find Corresponding Values of \(b\)
Using \(a + b = 69\):
- If \(a = 49\), then \(b = 20\).
- If \(a = 20\), then \(b = 49\).

### Step 11: Write the Final Answer
Thus, we can express the square root:
\[ \sqrt{69 + 28\sqrt{5}} = \sqrt{49} + \sqrt{20} = 7 + 2\sqrt{5} \]

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