To find the missing number in the quadratic equation \(x^2 - 14x + __ = 0\), where the solutions are given as \(6\) and \(8\), we can use Vieta's formulas. According to Vieta's formulas, for a quadratic equation in the form \(x^2 + bx + c = 0\):
- The sum of the roots (solutions) is given by \(-b\),
- The product of the roots is given by \(c\).
For the given equation, the sum of the roots \(6 + 8\) is \(14\), which corresponds to \(-(-14)\) in the equation. Therefore, this part is already correct.
Next, we need to calculate the product of the roots:
\[ 6 \times 8 = 48. \]
Since the product of the roots corresponds to the constant term (the term represented by the blank) in the quadratic equation, we determine that:
\[ c = 48. \]
Therefore, we can fill in the blank with \(48\).
The complete equation is:
\[ x^2 - 14x + 48 = 0. \]
Thus, the number that fills the blank is \(\boxed{48}\).