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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.\newlineg(x)=2x2−7x+5 g(x)=2x^{2}-7x+5 , g(x)=□(x+□)2+□ g(x)=\square(x+\square)^{2}+\square

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Q. Rewrite the function by completing the square.\newlineg(x)=2x2−7x+5 g(x)=2x^{2}-7x+5 , g(x)=□(x+□)2+□ g(x)=\square(x+\square)^{2}+\square
  1. Start with function: Start with the given function. g(x)=2x2−7x+5g(x) = 2x^2 - 7x + 5
  2. Factor out coefficient of x^22: Factor out the coefficient of x^22 from the first two terms.\newlineg(x) = 22(x^22 - (\frac{77}{22})x) + 55
  3. 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 xx in the parentheses.\newline(74)2=4916(\frac{7}{4})^2 = \frac{49}{16}\newlineAdd and subtract 4916\frac{49}{16} inside the parentheses.\newlineg(x)=2(x2−(72)x+4916−4916)+5g(x) = 2(x^2 - (\frac{7}{2})x + \frac{49}{16} - \frac{49}{16}) + 5
  4. Add and subtract inside parentheses: Add 4916\frac{49}{16} inside the parentheses and subtract its equivalent outside the parentheses to keep the equation balanced. The equivalent of adding 4916\frac{49}{16} inside the parentheses is adding 2×49162 \times \frac{49}{16} outside because of the factor of 22 in front of the parentheses.\newlineg(x) = 2(x2−(72)x+4916)−2×4916+52\left(x^2 - \left(\frac{7}{2}\right)x + \frac{49}{16}\right) - 2 \times \frac{49}{16} + 5
  5. Simplify equation by combining like terms: Simplify the equation by combining like terms outside the parentheses.\newlineg(x) = 22(x - \frac{77}{44})^22 - \frac{9898}{1616} + 55\newlineg(x) = 22(x - \frac{77}{44})^22 - \frac{9898}{1616} + \frac{8080}{1616}\newlineg(x) = 22(x - \frac{77}{44})^22 - \frac{1818}{1616}
  6. Simplify constant term: Simplify the constant term. g(x)=2(x−74)2−98g(x) = 2(x - \frac{7}{4})^2 - \frac{9}{8}

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