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:
- \(x - 7 = 0\) ⇒ \(x = 7\)
- \(x + 9 = 0\) ⇒ \(x = -9\)
So, the solutions are \(x = 7\) and \(x = -9\).
Now, let's analyze the given options to see which one has the same solutions.
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\(x^2 = 2x - 63\): 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.
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\(x^2 - 2x - 63 = 0\): 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 \(x = -9\).
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\(x^2 = -2x + 63\): 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.
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\(x^2 + 2x = -63\): 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:
\(x^2 = -2x + 63\).