Alright, let's solve the quadratic equation step-by-step to find its roots.
The given quadratic equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ a = 1, \quad b = -22, \quad c = 121 \][/tex]
We will use the quadratic formula to find the roots:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step-by-Step Solution:
1. Identify the coefficients:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = -22 \][/tex]
[tex]\[ c = 121 \][/tex]
2. Compute the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values:
[tex]\[ \Delta = (-22)^2 - 4(1)(121) \][/tex]
[tex]\[ \Delta = 484 - 484 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
So, the discriminant [tex]\( \Delta \)[/tex] is 0.
3. Since the discriminant is zero, there is exactly one unique solution (a repeated root). We now compute this root:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values and discriminant:
[tex]\[ x = \frac{-(-22) \pm \sqrt{0}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{22 \pm 0}{2} \][/tex]
[tex]\[ x = \frac{22}{2} \][/tex]
[tex]\[ x = 11 \][/tex]
Therefore, we have a repeated root:
[tex]\[ x = 11 \][/tex]
### Summary:
- The discriminant ([tex]\(\Delta\)[/tex]) is 0.
- There is one unique solution (a repeated root), which is [tex]\( x = 11 \)[/tex].
So the two roots are:
[tex]\[
x_1 = 11 \quad \text{and} \quad x_2 = 11
\][/tex]
This matches the solution given.
