Which of the following equations has exactly the same solutions as (x−7)(x+9)=0 ?(1 point)

To find which equation has exactly the same solutions as \((x−7)(x+9)=0\), we first determine the solutions of the original equation.

1. Set the factors equal to zero:
- \(x - 7 = 0 \Rightarrow x = 7\)
- \(x + 9 = 0 \Rightarrow x = -9\)

The solutions to the equation \((x−7)(x+9)=0\) are \(x = 7\) and \(x = -9\).

Now, we will evaluate each of the provided equations to see if they yield the same solutions.

1. **Equation**: \(x^2 - 2x - 63 = 0\)

To solve for \(x\), we can factor or use the quadratic formula. Factoring directly:
- \((x - 9)(x + 7) = 0\)
- Solutions: \(x = 9\) and \(x = -7\) (not the same).

2. **Equation**: \(x^2 = -2x + 63\) (Rearranging gives \(x^2 + 2x - 63 = 0\))

This is the same as the previous one once rearranged:
- \((x - 7)(x + 9) = 0\)
- Solutions: \(x = 7\) and \(x = -9\) (this is correct).

3. **Equation**: \(x^2 = 2x - 63\) (Rearranging gives \(x^2 - 2x + 63 = 0\))

Factoring or using the quadratic formula:
- Solutions are complex (no real roots).

4. **Equation**: \(x^2 + 2x = -63\) (Rearranging gives \(x^2 + 2x + 63 = 0\))

Similarly, this quadratic has no real roots due to the negative discriminant.

From this analysis, the equation that has exactly the same solutions as \((x−7)(x+9) = 0\) is:

**x^2 = −2x + 63**.