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 equals negative 2 x plus 63

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

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

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 this equation.

Setting each factor equal to zero gives:

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

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

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

1. Option 1: \(x^2 = -2x + 63\)
   Rearranging gives:
   \(x^2 + 2x - 63 = 0\)
   Factoring:
   \((x + 9)(x - 7) = 0\)
   Solutions: \(x = 7\) and \(x = -9\) (same as the original equation).

2. Option 2: \(x^2 = 2x - 63\)
   Rearranging gives:
   \(x^2 - 2x + 63 = 0\)
   Using the quadratic formula:
   \[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1}\]
   The discriminant is negative, so there are no real solutions.

3. Option 3: \(x^2 - 2x - 63 = 0\)
   Factoring gives:
   \((x - 9)(x + 7) = 0\)
   Solutions: \(x = 9\) and \(x = -7\) (not the same).

4. Option 4: \(x^2 + 2x = -63\)
   Rearranging gives:
   \(x^2 + 2x + 63 = 0\)
   Using the quadratic formula:
   The discriminant is again negative, meaning there are no real solutions.

The only option that matches the original solutions \(x = 7\) and \(x = -9\) is:

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