Question

Which 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 x squared plus 2 x equals negative 63 x2=−2x+63 x squared equals negative 2 x plus 63 x2−2x−63=0

Answer 1

To find which equation has the same solutions as \((x−7)(x+9)=0\), we first solve the original equation:

\((x−7)(x+9)=0\)

This gives us the solutions:
1. \(x - 7 = 0 \Rightarrow x = 7\)
2. \(x + 9 = 0 \Rightarrow x = -9\)

Now, we will analyze the provided options to see which one produces the same solutions (i.e., \(x = 7\) and \(x = -9\)).

1. **Option:** \(x^2 = 2x - 63\)
Rearranging, we get \(x^2 - 2x + 63 = 0\). The roots of this equation do not match \(7\) or \(-9\).

2. **Option:** \(x^2 + 2x = -63\)
Rearranging, we get \(x^2 + 2x + 63 = 0\). The roots of this equation do not match \(7\) or \(-9\).

3. **Option:** \(x^2 = -2x + 63\)
Rearranging, we get \(x^2 + 2x - 63 = 0\). We can solve this using the quadratic formula or factoring, and it does indeed factor to give us:
\((x - 7)(x + 9) = 0\), which has the same solutions \(x = 7\) and \(x = -9\).

4. **Option:** \(x^2 = 2x + 63\)
Rearranging gives \(x^2 - 2x - 63 = 0\). Solving this yields solutions that do not match \(7\) or \(-9\).

5. **Option:** \(x^2 - 2x - 63 = 0\)
This option presents the same quadratic form as mentioned in the previous one, and thus it can be rewritten as \((x - 9)(x + 7) = 0\), which again gives us \(x = 7\) and \(x = -9\).

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

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