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

1 answer

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.