To find the roots of the quadratic equation \(x^2 - 7x + 12 = 0\), we will factorize the equation.

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Here, \(a = 1\), \(b = -7\), and \(c = 12\).

The equation can be factorized by looking for two numbers whose product is \(c = 12\) and whose sum is \(b = -7\).

1. Possible pairs of numbers that multiply to 12 are:
   • \((1, 12)\)
   • \((2, 6)\)
   • \((3, 4)\)
2. Checking these pairs:
   • \(1 + 12 = 13\)
   • \(2 + 6 = 8\)
   • \(3 + 4 = 7\)

Thus, the factors of the equation \(x^2 - 7x + 12\) can be written as:

\((x - 3)(x - 4) = 0\).

Setting each factor equal to zero gives the roots of the equation:

• \(x - 3 = 0 \Rightarrow x = 3\)
• \(x - 4 = 0 \Rightarrow x = 4\)

Hence, the roots of the equation are \({3, 4}\), which matches the given correct answer.