To complete the square for the polynomial \( g^2 + 2g \), we want to express it in the form of \((g + a)^2\).
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Start with the expression: \[ g^2 + 2g \]
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Take the coefficient of \( g \), which is 2, divide it by 2, and square it: \[ \left(\frac{2}{2}\right)^2 = 1 \]
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Now, add and subtract this number (1) inside the expression: \[ g^2 + 2g + 1 - 1 \]
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Now, rewrite the expression: \[ (g + 1)^2 - 1 \]
Thus, the number you need to fill in the blank to make \( g^2 + 2g \) a perfect-square quadratic is 1.
So, the completed polynomial is: \[ g^2 + 2g + 1 \] which is \((g + 1)^2\).
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