To find the value of \( k \) that solves the equation \(-x^2 + 5x = -x + 5\), we first need to rearrange the equation into a standard form. Let's start by moving all terms to one side of the equation:
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Start with the original equation: \[ -x^2 + 5x = -x + 5 \]
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Move \(-x\) and \(5\) to the left side: \[ -x^2 + 5x + x - 5 = 0 \]
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Combine like terms: \[ -x^2 + 6x - 5 = 0 \]
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Multiply the entire equation by \(-1\) to make the leading coefficient positive: \[ x^2 - 6x + 5 = 0 \]
Now, we can solve this quadratic equation using factoring. We need two numbers that multiply to \(5\) and add up to \(-6\).
The numbers that fit this criteria are \(-1\) and \(-5\): \[ (x - 1)(x - 5) = 0 \]
Setting each factor equal to zero gives the solutions: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
Therefore, the solutions to the equation \(-x^2 + 5x = -x + 5\) are \(x = 1\) and \(x = 5\).
If \(k\) refers to these solutions, then the possible values of \(k\) would be \(1\) and \(5\). If the table mentioned in your question includes the solutions or options regarding \(k\), check if either of these solutions is present.
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