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$Rewrite the function by completing the square. {:[g(x)=2x^(2)-7x+5],[g(x)=◻(x+◻)^(2)+◻]:}$

Rewrite the function by completing the square. $\newline$ $g(x)=2x^{2}-7x+5$ , $g(x)=\square(x+\square)^{2}+\square$

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Math Problems

Algebra 1

Solve a quadratic equation by completing the square

Full solution

Q. Rewrite the function by completing the square.

\newline

g(x)=2x^{2}-7x+5

g(x)=\square(x+\square)^{2}+\square

Start with function: Start with the given function. $g(x) = 2x^2 - 7x + 5$
Factor out coefficient of x^ $2$ : Factor out the coefficient of x^ $2$ from the first two terms. $\newline$ g(x) = $2$ (x^ $2$ - (\frac{ $7$ }{ $2$ })x) + $5$
Complete the square: To complete the square, find the value that needs to be added and subtracted inside the parentheses. This value is the square of half the coefficient of $x$ in the parentheses. $\newline$ $(\frac{7}{4})^2 = \frac{49}{16}$ $\newline$ Add and subtract $\frac{49}{16}$ inside the parentheses. $\newline$ $g(x) = 2(x^2 - (\frac{7}{2})x + \frac{49}{16} - \frac{49}{16}) + 5$
Add and subtract inside parentheses: Add $\frac{49}{16}$ inside the parentheses and subtract its equivalent outside the parentheses to keep the equation balanced. The equivalent of adding $\frac{49}{16}$ inside the parentheses is adding $2 \times \frac{49}{16}$ outside because of the factor of $2$ in front of the parentheses. $\newline$ g(x) = $2\left(x^2 - \left(\frac{7}{2}\right)x + \frac{49}{16}\right) - 2 \times \frac{49}{16} + 5$
Simplify equation by combining like terms: Simplify the equation by combining like terms outside the parentheses. $\newline$ g(x) = $2$ (x - \frac{ $7$ }{ $4$ })^ $2$ - \frac{ $98$ }{ $16$ } + $5$ $\newline$ g(x) = $2$ (x - \frac{ $7$ }{ $4$ })^ $2$ - \frac{ $98$ }{ $16$ } + \frac{ $80$ }{ $16$ } $\newline$ g(x) = $2$ (x - \frac{ $7$ }{ $4$ })^ $2$ - \frac{ $18$ }{ $16$ }
Simplify constant term: Simplify the constant term. $g(x) = 2(x - \frac{7}{4})^2 - \frac{9}{8}$