To find the number that completes the quadratic equation \( x^2 - 14x + \text{?} = 0 \) given that the solutions are 6 and 8, we can use the fact that for a quadratic equation of the form \( ax^2 + bx + c = 0 \), the solutions can be determined using the relationships provided by Vieta's formulas.
According to Vieta's formulas:
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)
For our quadratic equation:
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Sum of the roots:
Given roots are 6 and 8. \[ 6 + 8 = 14 \]This matches with \(-\frac{-14}{1}\) (since \(b = -14\)). So this condition is satisfied.
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Product of the roots:
The product of the roots is calculated as follows: \[ 6 \cdot 8 = 48 \]Now using Vieta's formula for the product: \[ \frac{c}{a} = \frac{c}{1} = c \] Therefore: \[ c = 48 \]
So the quadratic equation we need is: \[ x^2 - 14x + 48 = 0 \]
Thus, the number that completes the quadratic equation is 48.