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

Question

?(1 point)
Responses

x2=−2x+63
x squared equals negative 2 x plus 63

x2=2x−63
x squared equals 2 x minus 63

x2+2x=−63
x squared plus 2 x equals negative 63

x2−2x−63=0

Answer 1

To solve the equation \((x−7)(x+9)=0\), we can find the solutions by setting each factor equal to zero:

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

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

Now, we need to determine which of the given equations has the same solutions:

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

2. \(x^2 = 2x - 63\)
   Rearranging gives:
   \(x^2 - 2x + 63 = 0\)
   We can check for solutions, but it does not equal the original factors.

3. \(x^2 + 2x = -63\)
   Rearranging gives:
   \(x^2 + 2x + 63 = 0\)
   This is not factorable with real solutions, so it is not the same.

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

Thus, the correct choice that matches the original equation \((x−7)(x+9)=0\) is:

\(x^2 = -2x + 63\).