Question

The difference between the roots of the quadratic equation x2+x+c=0 is 6. Find c.

Answer 1

We have that
$x_{1} - x_{2} = 6$
Let y be the first root, then y - 6 will be the second one. Since both roots make the quadratic equation equal to 0, we have that:
$y^{2} + y + c = (y - 6)^{2} + (y - 6) +c$
Solve for y:
$y^{2} +y +c = y^{2} - 12y + 36 + y - 6 +c \\ y + 12y - y = 30 \\ y = \frac{5}{2}$
Plug in y = 5/2 into original quadratic equation and solve for c:
$( \frac{5}{2}) ^{2} + \frac{5}{2} +c = 0 \\ c= - \frac{5}{2} - \frac{25}{4} \\ c = - \frac{35}{4}$
So, the answer is c = -35/4