Question

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 plus 2 x equals negative 63

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

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

x2=−2x+63

Answer 1

To determine which of the given equations has the same solutions as \((x-7)(x+9)=0\), we first solve \((x-7)(x+9)=0\).

Setting each factor to zero gives us:

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

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

Now, let's evaluate each of the given options to see which one has exactly the same solutions.

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

   Rearranging gives: \[ x^2 + 2x + 63 = 0 \] The discriminant is \(2^2 - 4 \cdot 1 \cdot 63 = 4 - 252 = -248\), which means there are no real solutions.

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

   We can factor this as: \[ (x-9)(x+7) = 0 \] Setting each factor to zero gives us solutions \(x = 9\) and \(x = -7\), which are different from \(7\) and \(-9\).

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

   Rearranging gives: \[ x^2 - 2x + 63 = 0 \] The discriminant is \((-2)^2 - 4 \cdot 1 \cdot 63 = 4 - 252 = -248\), which means there are no real solutions.

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

   Rearranging gives: \[ x^2 + 2x - 63 = 0 \] We can factor this as: \[ (x + 9)(x - 7) = 0 \] Setting each factor to zero gives us solutions \(x = -9\) and \(x = 7\), which matches our original solutions.

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

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