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$Rewrite the function by completing the square. {:[h(x)=x^(2)+3x-18],[h(x)=◻(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $\begin{array}{l} h(x)=x^{2}+3 x-18 \\ h(x)=\square(x+\square)^{2}+\square \end{array}$

Full solution

Q. Rewrite the function by completing the square.

\newline

\begin{array}{l} h(x)=x^{2}+3 x-18 \\ h(x)=\square(x+\square)^{2}+\square \end{array}

Identify coefficients: Identify the quadratic and linear coefficients from the function $h(x)$ . $\newline$ In $h(x) = x^2 + 3x - 18$ , the quadratic coefficient is $1$ (the coefficient of $x^2$ ) and the linear coefficient is $3$ (the coefficient of $x$ ).
Complete the square: Divide the linear coefficient by $2$ and square the result to find the number to complete the square. $\newline$ $(\frac{3}{2})^2 = \frac{9}{4}$ $\newline$ This is the value that will be added and subtracted inside the square to complete it.
Rewrite the function: Rewrite the function by adding and subtracting $(\frac{3}{2})^2$ inside the expression. $\newline$ $h(x) = x^2 + 3x + (\frac{3}{2})^2 - (\frac{3}{2})^2 - 18$ $\newline$ $h(x) = x^2 + 3x + \frac{9}{4} - \frac{9}{4} - 18$
Combine constant terms: Combine the constant terms outside the square. $\newline$ $-\frac{9}{4} - 18$ can be combined by converting $18$ to a fraction with a denominator of $4$ : $18 = \frac{72}{4}$ $\newline$ $h(x) = x^2 + 3x + \frac{9}{4} - \frac{72}{4} - \frac{9}{4}$ $\newline$ $h(x) = x^2 + 3x + \frac{9}{4} - \frac{81}{4}$
Factor and simplify: Factor the perfect square trinomial and simplify the constant term. $\newline$ $h(x) = (x + \frac{3}{2})^2 - \frac{81}{4}$ $\newline$ This is the function $h(x)$ rewritten by completing the square.