To find the number that should replace the blank in the quadratic equation \( x^2 - 14x + __ = 0 \) so that the solutions are \( 6 \) and \( 8 \), we can use Vieta's formulas. According to Vieta's, the sum of the roots (solutions) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \text{Sum of roots} = -\frac{b}{a} \]
And the product of the roots is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
In our equation, \( a = 1 \) and \( b = -14 \). The roots are \( 6 \) and \( 8 \).
Calculate the sum of the roots:
\[ 6 + 8 = 14 \]
This confirms that the coefficient of \( x \) (-14) is correct.
Calculate the product of the roots:
\[ 6 \cdot 8 = 48 \]
In the equation \( x^2 - 14x + c = 0 \), \( c \) must be equal to the product of the roots. So, we have:
\[ c = 48 \]
Thus, the complete quadratic equation is:
\[ x^2 - 14x + 48 = 0 \]
So, the number that should replace the blank is 48.