To find the equation that has exactly the same solutions as \((x−7)(x+9)=0\), we can start by solving the original equation.
Setting the factors to zero, we have:
- \(x - 7 = 0 \Rightarrow x = 7\)
- \(x + 9 = 0 \Rightarrow x = -9\)
The solutions to \((x−7)(x+9)=0\) are \(x = 7\) and \(x = -9\).
Next, we can check each of the given equations to see if they yield the same solutions:
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For \(x^2 = -2x + 63\): \[ x^2 + 2x - 63 = 0 \quad \text{(Rearranging gives us this quadratic equation)} \] We can factor this: \[ (x + 9)(x - 7) = 0 \] Solutions are \(x = 7\) and \(x = -9\).
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For \(x^2 = 2x - 63\): \[ x^2 - 2x + 63 = 0 \] This does not factor nicely and needs the quadratic formula: \[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1} = \frac{2 \pm \sqrt{-244}}{2} \] This has imaginary solutions, not matching \(7\) and \(-9\).
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For \(x^2 - 2x - 63 = 0\): Factoring gives: \[ (x - 9)(x + 7) = 0 \] Solutions are \(x = 9\) and \(x = -7\), which do not match.
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For \(x^2 + 2x = -63\): \[ x^2 + 2x + 63 = 0 \] This also needs the quadratic formula: \[ x = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot 63}}{2 \cdot 1} = \frac{-2 \pm \sqrt{-244}}{2} \] Like before, this does not yield real solutions.
Therefore, the equation that has exactly the same solutions as \((x−7)(x+9)=0\) is:
\(x^2 = -2x + 63\).