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

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

The equation \((x-7)(x+9)=0\) is satisfied when either factor is equal to zero:

1. \(x - 7 = 0\) → \(x = 7\)
2. \(x + 9 = 0\) → \(x = -9\)

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

Now, let's analyze each of the given equations to see if they produce the same solutions.

1. Equation: \(x^2 = 2x - 63\)
   Rearranging gives:
   \(x^2 - 2x + 63 = 0\) (Does not have the same solutions; leads to different roots)

2. Equation: \(x^2 + 2x = -63\)
   Rearranging gives:
   \(x^2 + 2x + 63 = 0\) (Does not have the same solutions; leads to complex roots)

3. Equation: \(x^2 = -2x + 63\)
   Rearranging gives:
   \(x^2 + 2x - 63 = 0\)
   Factoring gives: \((x-7)(x+9) = 0\) (This has the same solutions: \(x = 7\) and \(x = -9\))

4. Equation: \(x^2 - 2x - 63 = 0\)
   Factoring gives: \((x-9)(x+7) = 0\) (This has solutions \(x = 9\) and \(x = -7\) which are different)

Thus, the equation that has exactly the same solutions as \((x-7)(x+9)=0\) is:

x^2 = -2x + 63