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