Question

Supply the number so that the quadratic equation below has 6 and 8 as its solutions.(1 point)%0D%0Ax2−14x+%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A14%0D%0A𝑥%0D%0A+%0D%0A=0

Answer 1

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:

1. **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.

2. **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**.