Answer :
To determine which of the given options has the same solutions as the equation [tex]\( x^2 - 10x - 11 = 0 \)[/tex], let's solve this quadratic equation first.
### Step 1: Solve the Quadratic Equation [tex]\( x^2 - 10x - 11 = 0 \)[/tex]
The quadratic formula is [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]; here, [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -11\)[/tex].
So,
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-11)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{100 + 44}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{144}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm 12}{2} \][/tex]
This results in two solutions:
[tex]\[ x = \frac{10 + 12}{2} = 11 \][/tex]
[tex]\[ x = \frac{10 - 12}{2} = -1 \][/tex]
Thus, the solutions to the equation [tex]\( x^2 - 10x - 11 = 0 \)[/tex] are [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex].
### Step 2: Check Each Option
Now, we need to determine which of the given options A, B, C, or D has the same solutions.
Option A: [tex]\((x - 5)^2 = 36\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 5)^2 = 36 \][/tex]
Take the square root of both sides:
[tex]\[ x - 5 = \pm 6 \][/tex]
This gives us two solutions:
[tex]\[ x - 5 = 6 \implies x = 11 \][/tex]
[tex]\[ x - 5 = -6 \implies x = -1 \][/tex]
Both [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex] match the solutions from the original equation. Thus, Option A is correct.
Option B: [tex]\((x - 5)^2 = 21\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 5)^2 = 21 \][/tex]
Take the square root of both sides:
[tex]\[ x - 5 = \pm \sqrt{21} \][/tex]
This gives us two solutions:
[tex]\[ x - 5 = \sqrt{21} \implies x = 5 + \sqrt{21} \][/tex]
[tex]\[ x - 5 = -\sqrt{21} \implies x = 5 - \sqrt{21} \][/tex]
These solutions do not match the solutions from the original equation. Therefore, Option B is not correct.
Option C: [tex]\((x - 10)^2 = 36\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 10)^2 = 36 \][/tex]
Take the square root of both sides:
[tex]\[ x - 10 = \pm 6 \][/tex]
This gives us two solutions:
[tex]\[ x - 10 = 6 \implies x = 16 \][/tex]
[tex]\[ x - 10 = -6 \implies x = 4 \][/tex]
These solutions do not match the solutions from the original equation. Therefore, Option C is not correct.
Option D: [tex]\((x - 10)^2 = 21\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 10)^2 = 21 \][/tex]
Take the square root of both sides:
[tex]\[ x - 10 = \pm \sqrt{21} \][/tex]
This gives us two solutions:
[tex]\[ x - 10 = \sqrt{21} \implies x = 10 + \sqrt{21} \][/tex]
[tex]\[ x - 10 = -\sqrt{21} \implies x = 10 - \sqrt{21} \][/tex]
These solutions do not match the solutions from the original equation. Therefore, Option D is not correct.
### Conclusion
The correct option is:
[tex]\[ \boxed{A} \][/tex]
### Step 1: Solve the Quadratic Equation [tex]\( x^2 - 10x - 11 = 0 \)[/tex]
The quadratic formula is [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]; here, [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -11\)[/tex].
So,
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot (-11)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{100 + 44}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{144}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm 12}{2} \][/tex]
This results in two solutions:
[tex]\[ x = \frac{10 + 12}{2} = 11 \][/tex]
[tex]\[ x = \frac{10 - 12}{2} = -1 \][/tex]
Thus, the solutions to the equation [tex]\( x^2 - 10x - 11 = 0 \)[/tex] are [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex].
### Step 2: Check Each Option
Now, we need to determine which of the given options A, B, C, or D has the same solutions.
Option A: [tex]\((x - 5)^2 = 36\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 5)^2 = 36 \][/tex]
Take the square root of both sides:
[tex]\[ x - 5 = \pm 6 \][/tex]
This gives us two solutions:
[tex]\[ x - 5 = 6 \implies x = 11 \][/tex]
[tex]\[ x - 5 = -6 \implies x = -1 \][/tex]
Both [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex] match the solutions from the original equation. Thus, Option A is correct.
Option B: [tex]\((x - 5)^2 = 21\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 5)^2 = 21 \][/tex]
Take the square root of both sides:
[tex]\[ x - 5 = \pm \sqrt{21} \][/tex]
This gives us two solutions:
[tex]\[ x - 5 = \sqrt{21} \implies x = 5 + \sqrt{21} \][/tex]
[tex]\[ x - 5 = -\sqrt{21} \implies x = 5 - \sqrt{21} \][/tex]
These solutions do not match the solutions from the original equation. Therefore, Option B is not correct.
Option C: [tex]\((x - 10)^2 = 36\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 10)^2 = 36 \][/tex]
Take the square root of both sides:
[tex]\[ x - 10 = \pm 6 \][/tex]
This gives us two solutions:
[tex]\[ x - 10 = 6 \implies x = 16 \][/tex]
[tex]\[ x - 10 = -6 \implies x = 4 \][/tex]
These solutions do not match the solutions from the original equation. Therefore, Option C is not correct.
Option D: [tex]\((x - 10)^2 = 21\)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 10)^2 = 21 \][/tex]
Take the square root of both sides:
[tex]\[ x - 10 = \pm \sqrt{21} \][/tex]
This gives us two solutions:
[tex]\[ x - 10 = \sqrt{21} \implies x = 10 + \sqrt{21} \][/tex]
[tex]\[ x - 10 = -\sqrt{21} \implies x = 10 - \sqrt{21} \][/tex]
These solutions do not match the solutions from the original equation. Therefore, Option D is not correct.
### Conclusion
The correct option is:
[tex]\[ \boxed{A} \][/tex]
