Rewrite the function by completing the square. g(x)=4x^2-28x+49 g(x)=square (x+square )^2+square

To rewrite the function \( g(x) = 4x^2 - 28x + 49 \) by completing the square, follow these steps:<br /><br />1. **Factor out the coefficient of \( x^2 \) from the first two terms:**<br /> \[<br /> g(x) = 4(x^2 - 7x) + 49<br /> \]<br /><br />2. **Complete the square inside the parentheses:**<br /> - Take half of the coefficient of \( x \), which is \(-7\), divide it by 2, and then square it:<br /> \[<br /> \left( \frac{-7}{2} \right)^2 = \frac{49}{4}<br /> \]<br /> - Add and subtract this square inside the parentheses:<br /> \[<br /> g(x) = 4 \left( x^2 - 7x + \frac{49}{4} - \frac{49}{4} \right) + 49<br /> \]<br /> - Simplify inside the parentheses:<br /> \[<br /> g(x) = 4 \left( \left( x - \frac{7}{2} \right)^2 - \frac{49}{4} \right) + 49<br /> \]<br /><br />3. **Distribute the 4 and simplify:**<br /> \[<br /> g(x) = 4 \left( x - \frac{7}{2} \right)^2 - 4 \cdot \frac{49}{4} + 49<br /> \]<br /> \[<br /> g(x) = 4 \left( x - \frac{7}{2} \right)^2 - 49 + 49<br /> \]<br /> \[<br /> g(x) = 4 \left( x - \frac{7}{2} \right)^2<br /> \]<br /><br />So, the function \( g(x) \) rewritten by completing the square is:<br />\[<br />g(x) = 4 \left( x - \frac{7}{2} \right)^2<br />\]<br /><br />Therefore, the final answer is:<br />\[<br />g(x) = 4 \left( x - \frac{7}{2} \right)^2<br />\]