What is the value of `sqrt(34-2sqrt(253))` ?

Options

A. `sqrt(23) - sqrt(11)`
B. `sqrt(15) - sqrt(9)`
C. `sqrt(14) + sqrt(10)`
D. `sqrt(17) - sqrt(7)`

Answer

To find the value of \( \sqrt{34 - 2\sqrt{253}} \), we can follow these steps:

### Step 1: Identify the expression
We have the expression \( \sqrt{34 - 2\sqrt{253}} \).

### Step 2: Rewrite the expression
We can express \( 34 - 2\sqrt{253} \) in the form of \( (\sqrt{a} - \sqrt{b})^2 \). This means we need to find \( a \) and \( b \) such that:
\[ (\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab} \]
Here, we want \( a + b = 34 \) and \( 2\sqrt{ab} = 2\sqrt{253} \).

### Step 3: Solve for \( ab \)
From \( 2\sqrt{ab} = 2\sqrt{253} \), we can simplify to:
\[ \sqrt{ab} = \sqrt{253} \]
Squaring both sides gives:
\[ ab = 253 \]

### Step 4: Set up the equations
Now we have two equations:
1. \( a + b = 34 \)
2. \( ab = 253 \)

### Step 5: Solve the equations
We can treat \( a \) and \( b \) as the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \]
Substituting our values gives:
\[ x^2 - 34x + 253 = 0 \]

### Step 6: Use the quadratic formula
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation:
- \( a = 1 \)
- \( b = -34 \)
- \( c = 253 \)

Calculating the discriminant:
\[ b^2 - 4ac = 34^2 - 4 \cdot 1 \cdot 253 = 1156 - 1012 = 144 \]
Now substituting into the formula:
\[ x = \frac{34 \pm \sqrt{144}}{2} = \frac{34 \pm 12}{2} \]
This gives us two solutions:
\[ x = \frac{46}{2} = 23 \quad \text{and} \quad x = \frac{22}{2} = 11 \]
Thus, \( a = 23 \) and \( b = 11 \).

### Step 7: Rewrite the original expression
Now we can rewrite the original expression:
\[ \sqrt{34 - 2\sqrt{253}} = \sqrt{(\sqrt{23} - \sqrt{11})^2} \]
This simplifies to:
\[ \sqrt{34 - 2\sqrt{253}} = \sqrt{23} - \sqrt{11} \]

### Step 8: Final value
Thus, the value of \( \sqrt{34 - 2\sqrt{253}} \) is:
\[ \sqrt{23} - \sqrt{11} \]