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$Rewrite the function by completing the square. {:[h(x)=4x^(2)-36 x+81],[h(x)=◻(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $\begin{array}{l} h(x)=4 x^{2}-36 x+81 \\ h(x)=\square(x+\square)^{2}+\square \end{array}$

Full solution

Q. Rewrite the function by completing the square.

\newline

\begin{array}{l} h(x)=4 x^{2}-36 x+81 \\ h(x)=\square(x+\square)^{2}+\square \end{array}

Identify quadratic function: Identify the quadratic function to be rewritten. $h(x) = 4x^2 - 36x + 81$
Factor out coefficient of x^$2$: Factor out the coefficient of x^$2$ from the first two terms. $\newline$ h(x) = $4$ (x^$2$ - $9$ x) + $81$
Find value to complete the square: Find the value to complete the square. This is $(\frac{b}{2})^2$ , where $b$ is the coefficient of $x$ in the parentheses. $\newline$ For $x^2 - 9x$ , $b = -9$ , so $(\frac{b}{2})^2 = (\frac{-9}{2})^2 = \frac{81}{4}$ .
Add and subtract to complete the square: Add and subtract the value found in the previous step inside the parentheses to complete the square. $\newline$ $h(x) = 4(x^2 - 9x + \frac{81}{4} - \frac{81}{4}) + 81$
Rewrite trinomial as perfect square: Rewrite the trinomial as a perfect square and simplify the expression. $\newline$ $h(x) = 4\left(\left(x - \frac{9}{2}\right)^2 - \frac{81}{4}\right) + 81$
Distribute and combine like terms: Distribute the $4$ into the parentheses and combine like terms. $\newline$ $h(x) = 4(x - \frac{9}{2})^2 - 4(\frac{81}{4}) + 81$ $\newline$ $h(x) = 4(x - \frac{9}{2})^2 - 81 + 81$
Simplify constant terms: Simplify the constant terms. $\newline$ $h(x) = 4(x - \frac{9}{2})^2 + 0$ $\newline$ $h(x) = 4(x - \frac{9}{2})^2$