Question

Supply the number so that the quadratic equation has 6 and 8 as its solutions: x^2 - 14x + ___ = 0. The ___ stands for the blank you need to fill in.

Answer 1

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 \(\boxed{48}\).