To solve the quadratic equation [tex]\( x^2 + 9x - 9 = 0 \)[/tex] and express the solution in the form [tex]\( \frac{-9 \pm \sqrt{r}}{2} \)[/tex], we'll need to determine the value of [tex]\( r \)[/tex], where [tex]\( r \)[/tex] is the discriminant of the quadratic equation.
Given a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], the solution is derived using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this case, the coefficients are:
[tex]\[ a = 1, \; b = 9, \; c = -9 \][/tex]
The discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[ \Delta = 9^2 - 4 \cdot 1 \cdot (-9) \][/tex]
[tex]\[ \Delta = 81 + 36 \][/tex]
[tex]\[ \Delta = 117 \][/tex]
Therefore, in the expression [tex]\( \frac{-9 \pm \sqrt{r}}{2} \)[/tex], the value of [tex]\( r \)[/tex] is:
[tex]\[ r = 117 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{117} \][/tex]
