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

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

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Q. Rewrite the function by completing the square.

\newline

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

Divide and factor out leading coefficient: Divide the quadratic and linear coefficients by the leading coefficient if it is not $1$ . $\newline$ Since the leading coefficient of $f(x) = 2x^2 + 3x - 2$ is $2$ , we need to factor it out from the $x^2$ and $x$ terms. $\newline$ $f(x) = 2(x^2 + \frac{3}{2}x) - 2$
Complete the square: Find the value to complete the square. $\newline$ To complete the square for the expression $x^2 + \frac{3}{2}x$ , we need to add and subtract the square of half the coefficient of $x$ , which is $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$ . $\newline$ $f(x) = 2\left(x^2 + \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}\right) - 2$
Rewrite with perfect square trinomial: Rewrite the function with a perfect square trinomial. $\newline$ Add $\frac{9}{16}$ inside the parentheses and subtract $2 \times \frac{9}{16}$ (since we factored out a $2$ at the beginning) outside the parentheses to keep the equation balanced. $\newline$ $f(x) = 2\left(\left(x^2 + \frac{3}{2}x + \frac{9}{16}\right) - \frac{9}{16}\right) - 2$ $\newline$ $f(x) = 2\left(\left(x + \frac{3}{4}\right)^2 - \frac{9}{16}\right) - 2$
Distribute and simplify: Distribute the $2$ and simplify the constant terms. $f(x) = 2(x + \frac{3}{4})^2 - 2 \cdot \frac{9}{16} - 2$ $f(x) = 2(x + \frac{3}{4})^2 - \frac{9}{8} - \frac{16}{8}$ $f(x) = 2(x + \frac{3}{4})^2 - \frac{25}{8}$