To find which equation has exactly the same solutions as \((x - 7)(x + 9) = 0\), we first need to identify the solutions of the given equation.
- Set each factor to zero:
- \(x - 7 = 0\) gives \(x = 7\).
- \(x + 9 = 0\) gives \(x = -9\).
Thus, the solutions of the equation \((x - 7)(x + 9) = 0\) are \(x = 7\) and \(x = -9\).
Now, we can write a new equation using these solutions. The factors corresponding to the solutions can be expressed as: \[ (x - 7)(x + 9) = 0. \]
The original equation is already in factored form. However, we can also express it in standard quadratic form by expanding it: \[ x^2 + 9x - 7x - 63 = 0, \] which simplifies to: \[ x^2 + 2x - 63 = 0. \]
Any equation that can be expressed in the form of \((x - 7)(x + 9) = 0\) or that is equivalent to \(x^2 + 2x - 63 = 0\) will have the same solutions \(x = 7\) and \(x = -9\).
To summarize, any equation that is equivalent to \((x - 7)(x + 9) = 0\) or has the same roots will also have exactly these same solutions. Possible equivalent forms include:
- \((x - 7)(x + 9) = 0\)
- \(x^2 + 2x - 63 = 0\)
To check other equations, you would factor them or compute their roots to see if they yield \(x = 7\) and \(x = -9\) as solutions.
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