To complete the square for the expression \( r^2 + 24r \), we need to find the number that turns this into a perfect-square quadratic.
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Take the coefficient of \( r \), which is 24, divide it by 2, and then square the result: \[ \left(\frac{24}{2}\right)^2 = 12^2 = 144 \]
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Add this number (144) to the expression: \[ r^2 + 24r + 144 \]
Now, \( r^2 + 24r + 144 \) can be expressed as a perfect square: \[ (r + 12)^2 \]
Therefore, the number that makes the polynomial a perfect-square quadratic is 144.
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