Question

Which of the following has exactly the same solutions as (x – 7)(x + 9) = 0?

A. x^2 + 2x = -63
B. x^2 – 2x – 63 = 0
C. x^2 = -2x + 63
D. x^2 = 2x – 63

Answer 1

To determine which of the options has exactly the same solutions as the equation \((x - 7)(x + 9) = 0\), we first find the solutions to this equation.

Setting the factors equal to zero gives:

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

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

Now, we will check each option to see if they yield the same solutions.

**Option A:** \(x^2 + 2x = -63\)

Rearranging it gives:
\[
x^2 + 2x + 63 = 0
\]
We can use the quadratic formula to find the roots:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 252}}{2} = \frac{-2 \pm \sqrt{-248}}{2}
\]
This yields complex solutions, so this is not a match.

**Option B:** \(x^2 - 2x - 63 = 0\)

Factoring gives:
\[
(x - 9)(x + 7) = 0
\]
This results in:
1. \(x - 9 = 0\) gives \(x = 9\)
2. \(x + 7 = 0\) gives \(x = -7\)

These solutions are \(x = 9\) and \(x = -7\), so this is not a match.

**Option C:** \(x^2 = -2x + 63\)

Rearranging gives:
\[
x^2 + 2x - 63 = 0
\]
Factoring gives:
\[
(x - 7)(x + 9) = 0
\]
This yields the solutions \(x = 7\) and \(x = -9\), which are indeed the same as our original equation.

**Option D:** \(x^2 = 2x - 63\)

Rearranging gives:
\[
x^2 - 2x + 63 = 0
\]
We can check the discriminant:
\[
D = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot 63 = 4 - 252 = -248
\]
This gives complex solutions, so this is not a match.

Therefore, the option that has exactly the same solutions as \((x - 7)(x + 9) = 0\) is:

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