To complete the square for the equation \(2x^2 - 4x = 14\), follow these steps:
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Move the constant to the other side: \[ 2x^2 - 4x - 14 = 0 \]
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Factor out the coefficient of \(x^2\) (which is 2 in this case) from the terms on the left side: \[ 2(x^2 - 2x) = 14 \]
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Divide both sides by 2 to simplify: \[ x^2 - 2x = 7 \]
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Complete the square for the expression \(x^2 - 2x\):
- Take half of the coefficient of \(x\), which is \(-2\), giving \(-1\).
- Square this value: \((-1)^2 = 1\).
- Add and subtract this square inside the equation: \[ x^2 - 2x + 1 - 1 = 7 \] or \[ (x^2 - 2x + 1) - 1 = 7 \]
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Rewrite the left side as a square: \[ (x - 1)^2 - 1 = 7 \]
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Add 1 to both sides: \[ (x - 1)^2 = 8 \]
Now the equation is in completed square form: \[ (x - 1)^2 = 8 \]