Find the equation whose roots are 3 times the roots of the equation `x^(2)-5x+6=0`

Answer

To find the equation whose roots are three times the roots of the equation \(x^2 - 5x + 6 = 0\), we can follow these steps:

### Step 1: Identify the coefficients of the given quadratic equation
The given equation is:
\[ x^2 - 5x + 6 = 0 \]
Here, we identify:
- \(a = 1\)
- \(b = -5\)
- \(c = 6\)

### Step 2: Calculate the sum and product of the roots of the given equation
Let the roots of the equation be \(\alpha\) and \(\beta\).

Using Vieta's formulas:
- The sum of the roots \(\alpha + \beta\) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-5}{1} = 5 \]

- The product of the roots \(\alpha \beta\) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{6}{1} = 6 \]

### Step 3: Determine the new roots
The new roots are three times the original roots, so:
- New roots are \(3\alpha\) and \(3\beta\).

### Step 4: Calculate the sum and product of the new roots
- The sum of the new roots \(3\alpha + 3\beta\) is:
\[ 3(\alpha + \beta) = 3 \times 5 = 15 \]

- The product of the new roots \(3\alpha \cdot 3\beta\) is:
\[ (3\alpha)(3\beta) = 9(\alpha \beta) = 9 \times 6 = 54 \]

### Step 5: Form the new quadratic equation
The new quadratic equation can be formed using the sum and product of the new roots:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - 15x + 54 = 0 \]

### Final Answer
Thus, the equation whose roots are three times the roots of the given equation is:
\[ \boxed{x^2 - 15x + 54 = 0} \]