To solve for the value of [tex]\( r \)[/tex] in the expression [tex]\(\frac{11 \pm \sqrt{r}}{2}\)[/tex] from the quadratic equation [tex]\( x^2 - 11x + 5 \)[/tex], let's follow these steps:
1. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the quadratic equation:
[tex]\[
a = 1, \quad b = -11, \quad c = 5
\][/tex]
2. Calculate the discriminant of the quadratic equation:
The discriminant formula for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
3. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[
\text{Discriminant} = (-11)^2 - 4 \cdot 1 \cdot 5
\][/tex]
[tex]\[
\text{Discriminant} = 121 - 20
\][/tex]
[tex]\[
\text{Discriminant} = 101
\][/tex]
4. Interpret the discriminant in the context of the original question:
The question states that the solution is expressed as [tex]\(\frac{11 \pm \sqrt{r}}{2}\)[/tex]. This corresponds to the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex].
Since the discriminant [tex]\( b^2 - 4ac \)[/tex] is what is under the square root in the quadratic formula, we equate this to [tex]\( r \)[/tex]. Therefore:
[tex]\[
r = 101
\][/tex]
Therefore, the value of [tex]\( r \)[/tex] is [tex]\( \boxed{101} \)[/tex].
