X^2-7x-18=0
x^2+2x-9x-18=0
x(x+2)-9(x+2)=0
(x-9)(x+2)=0

soo roots are 9 and -2

Numerical difference =9-(-2)=11

Introduction:
To solve for the numerical difference of the roots of the equation x^2-7x-18=0, we first need to find the roots of the equation.

Finding the Roots:
To find the roots of the equation, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

For the equation x^2-7x-18=0, we have:

a = 1, b = -7, and c = -18

Substituting these values into the quadratic formula, we get:

x = (-(-7) ± sqrt((-7)^2 - 4(1)(-18))) / 2(1)

Simplifying the expression, we get:

x = (7 ± sqrt(85)) / 2

Therefore, the roots of the equation are:

x1 = (7 + sqrt(85)) / 2

x2 = (7 - sqrt(85)) / 2

Calculating the Numerical Difference:
To calculate the numerical difference of the roots, we simply subtract one root from the other:

x1 - x2 = [(7 + sqrt(85)) / 2] - [(7 - sqrt(85)) / 2]

Simplifying the expression, we get:

x1 - x2 = sqrt(85)

Therefore, the numerical difference of the roots of the equation x^2-7x-18=0 is sqrt(85).

Conclusion:
The numerical difference of the roots of the equation x^2-7x-18=0 is sqrt(85), which we calculated by finding the roots of the equation and subtracting one from the other.