Question

Which of the following equations has exactly the same solutions as (x−7)(x+9)=0 ?(1 point)
Responses

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
x squared plus 2 x equals negative 63

x2=2x−63

Answer 1

To determine which of the given equations has exactly the same solutions as \((x - 7)(x + 9) = 0\), we first find the solutions of the original equation:

The solutions occur when either factor is equal to zero:
1. \(x - 7 = 0\) which gives \(x = 7\)
2. \(x + 9 = 0\) which gives \(x = -9\)

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

Now, let's analyze each of the provided equations:

1. \(x^2 - 2x - 63 = 0\):
- To find the roots, we can use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-63)}}{2 \cdot 1}
\]
\[
x = \frac{2 \pm \sqrt{4 + 252}}{2} = \frac{2 \pm \sqrt{256}}{2} = \frac{2 \pm 16}{2}
\]
This gives us \(x = 9\) and \(x = -7\). Thus, this equation does not have the same solutions.

2. \(x^2 = -2x + 63\):
- Rearranging gives \(x^2 + 2x - 63 = 0\). This is the same as the first equation. So, it has the same solutions as before: does not match.

3. \(x^2 + 2x = -63\):
- Rearranging gives \(x^2 + 2x + 63 = 0\).
- Using the quadratic formula again:
\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 252}}{2}
\]
Here, we have a negative discriminant, which means no real solutions. Thus, this does not match.

4. \(x^2 = 2x - 63\):
- Rearranging gives \(x^2 - 2x + 63 = 0\).
- Using the quadratic formula:
\[
x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-63)}}{2 \cdot 1}
\]
\[
x = \frac{2 \pm \sqrt{4 + 252}}{2} = \frac{2 \pm \sqrt{256}}{2} = \frac{2 \pm 16}{2}
\]
This gives us \(x = 9\) and \(x = -7\). Thus, this equation does not have the same solutions.

Upon analyzing all four equations, none of them provide the same solutions, but the correct transformations, \(x^2 - 2x - 63 = 0\) and \(x^2 + 2x + 63 = 0\), lead to an incorrect match.

Based on all calculations, it appears the closest would likely have been \(x^2 - 2x - 63 = 0\) or a mislabeling in the options.

Overall, none of the equations listed provide exactly the same solutions \(x = 7\) and \(x = -9\).