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 equals negative 2 x plus 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
x squared plus 2 x equals negative 63

Answer 1

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

The solutions occur when each factor is equal to zero:

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

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

Now, we will check each of the given equations to see if they yield the same solutions.

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

Rearranged: \(x^2 + 2x - 63 = 0\)

To factor it:
\[
(x + 9)(x - 7) = 0
\]
Solutions: \(x = 7\) and \(x = -9\) (Same solutions)

2. **Equation:** \(x^2 - 2x - 63 = 0\)

We can factor this:
\[
(x - 9)(x + 7) = 0
\]
Solutions: \(x = 9\) and \(x = -7\) (Different solutions)

3. **Equation:** \(x^2 = 2x - 63\)

Rearranged: \(x^2 - 2x + 63 = 0\)
The discriminant is negative, so there are no real solutions. (Different solutions)

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

Rearranged: \(x^2 + 2x + 63 = 0\)
The discriminant is also negative, so there are also no real solutions. (Different solutions)

The only equation that has exactly the same solutions as \((x−7)(x+9)=0\) is:

**x^2 = -2x + 63** (which simplifies to \(x^2 + 2x - 63 = 0\)).