Square root of `139 - 80sqrt3` is ?

Options

A. `5sqrt3+8`
B. `5sqrt3-8`
C. `5sqrt3+12`
D. `5sqrt3-12`

Answer

To find the square root of \( 139 - 80\sqrt{3} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{139 - 80\sqrt{3}} \]

### 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 positive numbers.

### Step 3: Square both sides
Squaring both sides gives us:
\[ 139 - 80\sqrt{3} = a + b - 2\sqrt{ab} \]

### Step 4: Equate the rational and irrational parts
From the equation, we can equate the rational and irrational parts:
1. \( a + b = 139 \)
2. \( -2\sqrt{ab} = -80\sqrt{3} \)

From the second equation, we can simplify:
\[ 2\sqrt{ab} = 80\sqrt{3} \implies \sqrt{ab} = 40\sqrt{3} \implies ab = 4800 \]

### Step 5: Solve the system of equations
Now we have a system of equations:
1. \( a + b = 139 \)
2. \( ab = 4800 \)

We can solve this system by substituting \( b = 139 - a \) into the second equation:
\[ a(139 - a) = 4800 \]
This simplifies to:
\[ 139a - a^2 = 4800 \implies a^2 - 139a + 4800 = 0 \]

### Step 6: Use the quadratic formula
We can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( A = 1, B = -139, C = 4800 \):
\[ a = \frac{139 \pm \sqrt{(-139)^2 - 4 \cdot 1 \cdot 4800}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-139)^2 = 19321 \]
\[ 4 \cdot 1 \cdot 4800 = 19200 \]
\[ 19321 - 19200 = 121 \]
Thus, we have:
\[ a = \frac{139 \pm \sqrt{121}}{2} = \frac{139 \pm 11}{2} \]

### Step 7: Find the values of \( a \) and \( b \)
Calculating the two possible values:
1. \( a = \frac{150}{2} = 75 \)
2. \( a = \frac{128}{2} = 64 \)

So, we have \( a = 75 \) and \( b = 64 \) (or vice versa).

### Step 8: Write the square root
Thus, we can express the square root as:
\[ \sqrt{139 - 80\sqrt{3}} = \sqrt{75} - \sqrt{64} \]
Calculating the square roots:
\[ \sqrt{75} = 5\sqrt{3}, \quad \sqrt{64} = 8 \]
So we have:
\[ \sqrt{139 - 80\sqrt{3}} = 5\sqrt{3} - 8 \]

### Final Answer
The final answer is:
\[ \sqrt{139 - 80\sqrt{3}} = 5\sqrt{3} - 8 \]