To solve the equation \( x = -\frac{1}{2} x (x - 20) \), we start by simplifying it.
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Distribute the \(-\frac{1}{2} x\): \[ x = -\frac{1}{2} x^2 + 10x \]
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Rearranging the equation to bring all terms to one side gives: \[ \frac{1}{2} x^2 - 10x + x = 0 \] Simplifying further: \[ \frac{1}{2} x^2 - 9x = 0 \]
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Factor the equation: \[ x(\frac{1}{2} x - 9) = 0 \]
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Setting each factor to zero gives: \[ x = 0 \quad \text{or} \quad \frac{1}{2} x - 9 = 0 \]
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From the second factor, solve for \( x \): \[ \frac{1}{2} x = 9 \] \[ x = 18 \]
Therefore, the solutions are: \[ x = 0 \quad \text{or} \quad x = 18 \]
So, the values of \( x \) that satisfy the equation are \( x = 0 \) and \( x = 18 \).