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

Question

?(1 point)
Responses

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

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

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

x^2−2x−63=0

Answer 1

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

Setting each factor to zero gives us:

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

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

Next, we can analyze each of the given options to see which one has these solutions.

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

Rearranging gives \(x^2 - 2x + 63 = 0\), which does not have the same roots.

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

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

To find the roots of \(x^2 + 2x - 63 = 0\), we can factor: \[ x^2 + 2x - 63 = (x + 9)(x - 7) = 0. \] The solutions are \(x = -9\) and \(x = 7\), which match.

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

Rearranging gives \(x^2 + 2x + 63 = 0\), which does not have the same roots.

4. \(x^2 - 2x - 63 = 0\)

Factoring gives: \[ (x - 9)(x + 7) = 0. \] The solutions are \(x = 9\) and \(x = -7\), which do not match.

Conclusion:

The equation that has exactly the same solutions as \((x - 7)(x + 9) = 0\) is: \[ \boxed{x^2 + 2x - 63 = 0.} \]