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

Question

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 solve the equation \((x−7)(x+9)=0\), we set each factor to zero:

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

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

Next, we need to check which of the given equations has the same solutions.

1. For the equation: \(x^2 + 2x = -63\)

   Rearranging gives: \[x^2 + 2x + 63 = 0\] The discriminant \(\Delta = b^2 - 4ac = 2^2 - 4(1)(63) = 4 - 252 = -248\) (no real solutions).

2. For the equation: \(x^2 - 2x - 63 = 0\)

   Factorizing: \[x^2 - 2x - 63 = (x - 9)(x + 7) = 0\] Solutions are \(x = 9\) and \(x = -7\) (not the same).

3. For the equation: \(x^2 = 2x - 63\)

   Rearranging gives: \[x^2 - 2x + 63 = 0\] The discriminant \(\Delta = (-2)^2 - 4(1)(63) = 4 - 252 = -248\) (no real solutions).

4. For the equation: \(x^2 = -2x + 63\)

   Rearranging gives: \[x^2 + 2x - 63 = 0\] Factorizing: \[(x - 7)(x + 9) = 0\] Solutions are \(x = 7\) and \(x = -9\) (the same as the original equation).

Therefore, the equation that has exactly the same solutions as \((x−7)(x+9)=0\) is: \[x^2 = -2x + 63\]