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

Question

?(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 equation has the same solutions as \((x - 7)(x + 9) = 0\), we first find the solutions to the given equation.

Setting each factor to zero, we have:

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

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

Now, we need to check each of the given options to see which one has the same set of solutions.

1. Option 1: \(x^2 + 2x = -63\) \[ x^2 + 2x + 63 = 0 \quad \text{(Rearranging)} \] This is a quadratic equation, and we need to check its discriminant: \[ D = b^2 - 4ac = (2)^2 - 4(1)(63) = 4 - 252 = -248 \quad \text{(No real solutions)} \]

2. Option 2: \(x^2 - 2x - 63 = 0\) We can factor this quadratic: \[ (x - 9)(x + 7) = 0 \] The solutions are: \[ x = 9 \quad \text{and} \quad x = -7 \quad \text{(not the same)} \]

3. Option 3: \(x^2 = 2x - 63\) Rearranging gives: \[ x^2 - 2x + 63 = 0 \] Again checking the discriminant: \[ D = (-2)^2 - 4(1)(63) = 4 - 252 = -248 \quad \text{(No real solutions)} \]

4. Option 4: \(x^2 = -2x + 63\) Rearranging gives: \[ x^2 + 2x - 63 = 0 \] We can factor this quadratic: \[ (x - 7)(x + 9) = 0 \] The solutions are: \[ x = 7 \quad \text{and} \quad x = -9 \quad \text{(same solutions)} \]

Thus, the correct response is:

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