Question

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

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

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

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

x2=2x−63

Answer 1

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

The solutions of \((x - 7)(x + 9) = 0\) can be found by setting each factor to zero:

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

Now, we can express the solutions \(x = 7\) and \(x = -9\) as roots of a quadratic equation. The corresponding quadratic equation is:

\[
(x - 7)(x + 9) = 0 \implies x^2 + 2x - 63 = 0.
\]

Next, let's identify the option that simplifies to this quadratic equation or gives the same solutions.

- **Option 1**: \(x^2 - 2x - 63 = 0\) has different roots.
- **Option 2**: \(x^2 = -2x + 63\) can be rearranged to \(x^2 + 2x - 63 = 0\).
- **Option 3**: \(x^2 + 2x = -63\) can also be rearranged to \(x^2 + 2x + 63 = 0\) which is incorrect.
- **Option 4**: \(x^2 = 2x - 63\) rearranges to \(x^2 - 2x + 63 = 0\) which also has different roots.

The only option that gives the same solutions as the original equation \((x - 7)(x + 9) = 0\) is:

**x² = -2x + 63**, which rearranges to become \(x^2 + 2x - 63 = 0\).

So based on the analysis, the correct answer is:

**x² = -2x + 63**