To complete the square for the equation \(4x^2 - 12x + 9 = 0\), we can follow these steps:
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Start by factoring out the coefficient of \(x^2\) from the first two terms. Here, the coefficient is 4: \[ 4(x^2 - 3x) + 9 = 0 \]
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Now, we complete the square inside the parentheses. To do this, take the coefficient of \(x\) (which is -3), divide it by 2, and square it: \[ \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \]
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Add and subtract \(\frac{9}{4}\) inside the parentheses: \[ 4\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) + 9 = 0 \]
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Rearranging gives: \[ 4\left((x - \frac{3}{2})^2 - \frac{9}{4}\right) + 9 = 0 \]
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Now distribute the 4: \[ 4(x - \frac{3}{2})^2 - 9 + 9 = 0 \]
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This simplifies to: \[ 4(x - \frac{3}{2})^2 = 0 \]
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Now, we can write the equation in completed square form: \[ \left(x - \frac{3}{2}\right)^2 = 0 \]
So the completed square form of the equation is: \[ \left(x - \frac{3}{2}\right)^2 = 0 \]
In the original requested format: \[ (x + \quad (-\frac{3}{2}))^2 = 0 \]
Thus, you can fill in:
- \( -\frac{3}{2} \)
- \( 0 \)
Final answer: \[ (x - \frac{3}{2})^2 = 0 \]
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