Find the equation whose roots are greater by 2 of the roots of `x^(2)-7x+12=0`

Answer

To find the equation whose roots are greater by 2 than the roots of the equation \(x^2 - 7x + 12 = 0\), we can follow these steps:

### Step 1: Find the roots of the given quadratic equation
The given equation is:
\[ x^2 - 7x + 12 = 0 \]
We can factor this equation. We look for two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of \(x\)).

The numbers -3 and -4 satisfy these conditions:
\[ x^2 - 3x - 4x + 12 = 0 \]
This can be factored as:
\[ (x - 3)(x - 4) = 0 \]
Thus, the roots are:
\[ x = 3 \quad \text{and} \quad x = 4 \]

### Step 2: Determine the new roots
The problem states that we need to find the roots that are greater by 2 than the original roots. Therefore, we add 2 to each of the roots:
\[ \text{New root 1} = 3 + 2 = 5 \]
\[ \text{New root 2} = 4 + 2 = 6 \]

### Step 3: Calculate the sum and product of the new roots
Now, we calculate the sum and product of the new roots:
\[ \text{Sum of new roots} = 5 + 6 = 11 \]
\[ \text{Product of new roots} = 5 \times 6 = 30 \]

### Step 4: Form the new quadratic equation
Using the sum and product of the roots, we can form the new quadratic equation using the standard form:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - 11x + 30 = 0 \]

### Final Answer
Thus, the equation whose roots are greater by 2 than the roots of the original equation is:
\[ \boxed{x^2 - 11x + 30 = 0} \]