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 2 x minus 63

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

x2=−2x+63
x squared equals negative 2 x plus 63

x2+2x=−63

Answer 1

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

The equation \((x−7)(x+9)=0\) is satisfied when either factor is equal to zero:

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

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

Now, let's check each option to see which one has the same solutions.

1. **Option 1: \(x^2 = 2x - 63\)**
Rearranging gives us:
\(x^2 - 2x + 63 = 0\)
We can use the quadratic formula to find the roots:
\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 252}}{2}\]
Since the discriminant \(4 - 252 < 0\), there are no real solutions.

2. **Option 2: \(x^2 - 2x - 63 = 0\)**
We can factor this:
\((x - 9)(x + 7) = 0\)
This gives us the solutions \(x = 9\) and \(x = -7\), which do not match.

3. **Option 3: \(x^2 = -2x + 63\)**
Rearranging gives us:
\(x^2 + 2x - 63 = 0\)
We can factor this:
\((x - 7)(x + 9) = 0\)
This gives us the solutions \(x = 7\) and \(x = -9\), which match!

4. **Option 4: \(x^2 + 2x = -63\)**
Rearranging gives us:
\(x^2 + 2x + 63 = 0\)
The discriminant is \(2^2 - 4 \cdot 1 \cdot 63 = 4 - 252 < 0\), so there are no real solutions.

Based on the above analyses, the equation with exactly the same solutions as \((x−7)(x+9)=0\) is:

**Option 3: \(x^2 + 2x - 63 = 0\)**.