Which of the following equations has exactly the same solutions as (x−7)(x+9)=0 ?(1 point)%0D%0AResponses%0D%0A%0D%0Ax2−2x−63=0%0D%0Ax squared minus 2 x minus 63 equals 0%0D%0A%0D%0Ax2=−2x+63%0D%0Ax squared equals negative 2 x plus 63%0D%0A%0D%0Ax2=2x−63%0D%0Ax squared equals 2 x minus 63%0D%0A%0D%0Ax2+2x=−63

Answer 1

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

Setting \((x−7)(x+9)=0\) gives us:

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

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

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

1. \(x^2 - 2x - 63 = 0\)
   We can factor or use the quadratic formula to find the roots: \[ x^2 - 2x - 63 = (x - 9)(x + 7) = 0 \] Thus, the solutions are \(x = 9\) and \(x = -7\). Not the same.

2. \(x^2 = -2x + 63\)
   Rearranging gives: \[ x^2 + 2x - 63 = 0 \] Factoring: \[ (x - 7)(x + 9) = 0 \] So the solutions are \(x = 7\) and \(x = -9\). This one matches!

3. \(x^2 = 2x - 63\)
   Rearranging gives: \[ x^2 - 2x + 63 = 0 \] This doesn’t factor nicely and has different roots. Not the same.

4. \(x^2 + 2x = -63\)
   Rearranging gives: \[ x^2 + 2x + 63 = 0 \] This also doesn’t yield the same solutions upon solving.

After checking all the equations, the one that has exactly the same solutions as \((x−7)(x+9)=0\) is:

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