Question

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

Answer 1

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

The product \((x-7)(x+9)=0\) gives us two solutions:

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

To have an equation with the same solutions, we need another equation that also factors to yield the same roots.

One possible equation is:
\[
(x - 7)(x + 9) = k
\]
for any non-zero constant \(k\). A valid alternative form can also be achieved by manipulating the original equation, such as adding the same expression to both sides or multiplying the equation by a non-zero constant.

For example:
1. \((x - 7)(x + 9) = 0\)
2. \((x - 7)(x + 9) + 1 = 1\) has the same roots since the left-hand side will still be 0 at \(x = 7\) and \(x = -9\).

To summarize, any equation derived from \((x - 7)(x + 9) = 0\) through algebraic manipulation (as long as you do not change the roots) will have the same solutions.

If you provide specific options, I can help identify the correct equation among them, if any exist.