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

?(1 point)
Responses

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

x2−2x−63=0
x squared minus 2 x minus 63 equals 0

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

x2+2x=−63

1 answer

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 to this equation.

The equation \((x - 7)(x + 9) = 0\) implies:

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

So, the solutions are \(x = 7\) and \(x = -9\).

Now, let's analyze the given options to see which one has the same solutions.

  1. \(x^2 = 2x - 63\): Rearranging gives \(x^2 - 2x + 63 = 0\). The discriminant \(D = b^2 - 4ac = (-2)^2 - 4163 = 4 - 252 = -248\), which means there are no real solutions.

  2. \(x^2 - 2x - 63 = 0\): This can be factored as \((x - 9)(x + 7) = 0\), giving solutions \(x = 9\) and \(x = -7\), which are not the same as \(x = 7\) and \(x = -9\).

  3. \(x^2 = -2x + 63\): Rearranging gives \(x^2 + 2x - 63 = 0\). This can be factored as \((x + 9)(x - 7) = 0\), giving the solutions \(x = 7\) and \(x = -9\), which are indeed the same as the original equation's solutions.

  4. \(x^2 + 2x = -63\): Rearranging gives \(x^2 + 2x + 63 = 0\). The discriminant \(D = 2^2 - 4163 = 4 - 252 = -248\), which means there are no real solutions.

From the above analysis, the option that has exactly the same solutions as \((x - 7)(x + 9) = 0\) is:

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