Question

Determine which equation has the same solutions as the given equation.

x2 − 10x − 11 = 0

A.
(x − 10)2 = 36
B.
(x − 10)2 = 21
C.
(x − 5)2 = 21
D.
(x − 5)2 = 36

Answer 1

To determine which equation has the same solutions as the given equation \( x^2 - 10x - 11 = 0 \), we first need to solve the original equation for \( x \).

We can factor the quadratic equation \( x^2 - 10x - 11 = 0 \). We are looking for two numbers that multiply to \(-11\) and add to \(-10\). These numbers are \(-11\) and \(1\).

Thus, we can factor the equation as follows:

\[
(x - 11)(x + 1) = 0
\]

Setting each factor to zero gives us the solutions:

1. \( x - 11 = 0 \) ⇒ \( x = 11 \)
2. \( x + 1 = 0 \) ⇒ \( x = -1 \)

So, the solutions to the equation \( x^2 - 10x - 11 = 0 \) are \( x = 11 \) and \( x = -1 \).

Now, we will check each of the given options to see if they yield the same solutions:

### Option A: \( (x - 10)^2 = 36 \)

Taking the square root of both sides:

\[
x - 10 = 6 \quad \text{or} \quad x - 10 = -6
\]

Solving these:

1. \( x = 16 \)
2. \( x = 4 \)

### Option B: \( (x - 10)^2 = 21 \)

Taking the square root of both sides:

\[
x - 10 = \sqrt{21} \quad \text{or} \quad x - 10 = -\sqrt{21}
\]

Solving these:

1. \( x = 10 + \sqrt{21} \)
2. \( x = 10 - \sqrt{21} \)

### Option C: \( (x - 5)^2 = 21 \)

Taking the square root of both sides:

\[
x - 5 = \sqrt{21} \quad \text{or} \quad x - 5 = -\sqrt{21}
\]

Solving these:

1. \( x = 5 + \sqrt{21} \)
2. \( x = 5 - \sqrt{21} \)

### Option D: \( (x - 5)^2 = 36 \)

Taking the square root of both sides:

\[
x - 5 = 6 \quad \text{or} \quad x - 5 = -6
\]

Solving these:

1. \( x = 11 \)
2. \( x = -1 \)

### Conclusion:

The correct option that has the same solutions \( x = 11 \) and \( x = -1 \) as the original equation is:

**D. \( (x - 5)^2 = 36 \)**.