What is the positive square root of `[19+4sqrt(21)]` ?

Options

A. `sqrt(7)+2sqrt(3)`
B. `sqrt(3)+2sqrt(7)`
C. `sqrt(2)+3sqrt(7)`
D. `sqrt(7)+3sqrt(3)`

Answer

To find the positive square root of the expression \( 19 + 4\sqrt{21} \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ x = \sqrt{19 + 4\sqrt{21}} \]

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

### Step 3: Square Both Sides
Squaring both sides gives us:
\[ x^2 = a + b + 2\sqrt{ab} \]
We know from our original expression that:
\[ x^2 = 19 + 4\sqrt{21} \]

### Step 4: Equate the Rational and Irrational Parts
From the equation \( a + b + 2\sqrt{ab} = 19 + 4\sqrt{21} \), we can equate the rational and irrational parts:
1. \( a + b = 19 \)
2. \( 2\sqrt{ab} = 4\sqrt{21} \)

### Step 5: Solve for \( ab \)
From the second equation:
\[ \sqrt{ab} = 2\sqrt{21} \]
Squaring both sides gives:
\[ ab = 4 \times 21 = 84 \]

### Step 6: Set Up a System of Equations
Now we have a system of equations:
1. \( a + b = 19 \)
2. \( ab = 84 \)

### Step 7: Use the Quadratic Formula
We can treat \( a \) and \( b \) as the roots of the quadratic equation:
\[ t^2 - (a+b)t + ab = 0 \]
Substituting the values we have:
\[ t^2 - 19t + 84 = 0 \]

### Step 8: Calculate the Discriminant
The discriminant \( D \) of this quadratic equation is:
\[ D = b^2 - 4ac = 19^2 - 4 \times 1 \times 84 = 361 - 336 = 25 \]

### Step 9: Find the Roots
Now we can find the roots using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{D}}{2a} = \frac{19 \pm 5}{2} \]
Calculating the two possible values:
1. \( t = \frac{24}{2} = 12 \)
2. \( t = \frac{14}{2} = 7 \)

Thus, \( a = 12 \) and \( b = 7 \) (or vice versa).

### Step 10: Write the Final Expression
Now substituting back, we have:
\[ \sqrt{19 + 4\sqrt{21}} = \sqrt{12} + \sqrt{7} \]
Since \( \sqrt{12} = 2\sqrt{3} \), we can write:
\[ \sqrt{19 + 4\sqrt{21}} = 2\sqrt{3} + \sqrt{7} \]

### Conclusion
The positive square root of \( 19 + 4\sqrt{21} \) is:
\[ \boxed{2\sqrt{3} + \sqrt{7}} \]