To find the expression \(2D - 3C^2\), we need to first determine \(D\) and \(C\), then calculate \(C^2\), and finally substitute these values into the expression.
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Given: \[ D = 1 + 4p - 6p^2 \]
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Given: \[ C = 1 - p \]
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First, calculate \(C^2\): \[ C^2 = (1 - p)^2 \] Expanding this using the formula \((a - b)^2 = a^2 - 2ab + b^2\): \[ C^2 = 1^2 - 2(1)(p) + p^2 = 1 - 2p + p^2 \]
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Next, calculate \(3C^2\): \[ 3C^2 = 3(1 - 2p + p^2) = 3 - 6p + 3p^2 \]
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Now calculate \(2D\): \[ 2D = 2(1 + 4p - 6p^2) = 2 + 8p - 12p^2 \]
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Now substitute \(2D\) and \(3C^2\) into the expression \(2D - 3C^2\): \[ 2D - 3C^2 = (2 + 8p - 12p^2) - (3 - 6p + 3p^2) \] Simplifying this: \[ = 2 + 8p - 12p^2 - 3 + 6p - 3p^2 \] \[ = (2 - 3) + (8p + 6p) + (-12p^2 - 3p^2) \] \[ = -1 + 14p - 15p^2 \]
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Finally, write the expression in standard form: \[ -15p^2 + 14p - 1 \]
Thus, the expression that equals \(2D - 3C^2\) in standard form is: \[ \boxed{-15p^2 + 14p - 1} \]