Question

Supply the number so that the quadratic equation below has 6 and 8 as it’s solution
x^2 + 14x + ____=0

Answer 1

To find the missing number in the quadratic equation \(x^2 + 14x + \_\_\_ = 0\) that has 6 and 8 as its solutions, we can use the fact that for a quadratic equation in the form \(ax^2 + bx + c = 0\), the solutions (roots) can be expressed in terms of the coefficients as follows:

1. The sum of the roots \(r_1 + r_2 = -\frac{b}{a}\)
2. The product of the roots \(r_1 \cdot r_2 = \frac{c}{a}\)

For our equation, we have:

- Roots: \(r_1 = 6\) and \(r_2 = 8\)
- Coefficient \(a = 1\) (since it's \(x^2\))
- Coefficient \(b = 14\)

**Sum of the roots:**

\[
6 + 8 = 14
\]

This matches directly with \(-b/a = -14/1\), confirming that \(b = 14\) is correct.

**Product of the roots:**

\[
6 \cdot 8 = 48
\]

Now, we can find \(c\) since \(c = a \cdot (r_1 \cdot r_2)\) and \(a = 1\):

\[
c = 1 \cdot 48 = 48
\]

Thus, the missing number is \(48\).

Putting it all together, the quadratic equation is:

\[
x^2 + 14x + 48 = 0
\]

So, the final answer is:

\[
\boxed{48}
\]