If one root of the quadratic equation `ax^(2) + bx + c = 0` is `15 + 2sqrt(56)` and `a,b` and c are rational, then find the quadratic equation.

Answer

To find the quadratic equation given that one root is \( 15 + 2\sqrt{56} \) and \( a, b, c \) are rational, we follow these steps:

### Step 1: Identify the conjugate root
Since one root is \( 15 + 2\sqrt{56} \) and the coefficients of the quadratic equation must be rational, the other root must be the conjugate of the first root. Thus, the second root is:
\[ 15 - 2\sqrt{56} \]

### Step 2: Use the roots to form the quadratic equation
The quadratic equation can be formed using the fact that if \( r_1 \) and \( r_2 \) are the roots, then the equation can be expressed as:
\[ (x - r_1)(x - r_2) = 0 \]
Substituting the roots:
\[ (x - (15 + 2\sqrt{56}))(x - (15 - 2\sqrt{56})) = 0 \]

### Step 3: Expand the equation
Using the difference of squares formula, we can simplify:
\[ (x - 15 - 2\sqrt{56})(x - 15 + 2\sqrt{56}) = (x - 15)^2 - (2\sqrt{56})^2 \]

### Step 4: Calculate each part
First, calculate \( (x - 15)^2 \):
\[ (x - 15)^2 = x^2 - 30x + 225 \]
Now calculate \( (2\sqrt{56})^2 \):
\[ (2\sqrt{56})^2 = 4 \times 56 = 224 \]

### Step 5: Combine the results
Now substituting back into the equation:
\[ (x - 15)^2 - (2\sqrt{56})^2 = x^2 - 30x + 225 - 224 \]
This simplifies to:
\[ x^2 - 30x + 1 = 0 \]

### Step 6: Write the final quadratic equation
Thus, the quadratic equation is:
\[ x^2 - 30x + 1 = 0 \]

### Summary
The quadratic equation is:
\[ x^2 - 30x + 1 = 0 \]